math.cos on complex, imaginary part

Percentage Accurate: 67.0% → 99.9%

Time: 10.0s

Alternatives: 19

Speedup: 1.5×

• 51.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 66.5

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

   7. lift--.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lift-neg.f64 N/A

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

   12. sinh-undef N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lift-sinh.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-def N/A

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

   14. lift-sinh.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\left(\sinh im \cdot -2\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -5e-306)
     (* (* (sinh im) -2.0) (* 0.5 re))
     (if (<= t_0 1e-7)
       (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -5e-306) {
		tmp = (sinh(im) * -2.0) * (0.5 * re);
	} else if (t_0 <= 1e-7) {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -5e-306)
		tmp = Float64(Float64(sinh(im) * -2.0) * Float64(0.5 * re));
	elseif (t_0 <= 1e-7)
		tmp = Float64(Float64(sin(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-306], N[(N[(N[Sinh[im], $MachinePrecision] * -2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.99999999999999998e-306

   1. Initial program 97.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 76.8

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

   7. Applied rewrites 76.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 76.8

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

   9. Applied rewrites 76.8%

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

   if -4.99999999999999998e-306 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 9.9999999999999995e-8

   1. Initial program 32.2%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-out N/A

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

   5. Applied rewrites 99.8%

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

   if 9.9999999999999995e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 77.7%

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

Alternative 3: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\left(\sinh im \cdot -2\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -5e-306)
     (* (* (sinh im) -2.0) (* 0.5 re))
     (if (<= t_0 1e-7)
       (* (- (sin re)) im)
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -5e-306) {
		tmp = (sinh(im) * -2.0) * (0.5 * re);
	} else if (t_0 <= 1e-7) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -5e-306)
		tmp = Float64(Float64(sinh(im) * -2.0) * Float64(0.5 * re));
	elseif (t_0 <= 1e-7)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-306], N[(N[(N[Sinh[im], $MachinePrecision] * -2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.99999999999999998e-306

   1. Initial program 97.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 76.8

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

   7. Applied rewrites 76.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 76.8

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

   9. Applied rewrites 76.8%

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

   if -4.99999999999999998e-306 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 9.9999999999999995e-8

   1. Initial program 32.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 9.9999999999999995e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 77.7%

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

Alternative 4: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, im \cdot im, 0.016666666666666666\right), im \cdot im, 0.3333333333333333\right), im \cdot im, 2\right) \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -5e-306)
     (*
      (-
       (*
        (fma
         (fma
          (fma 0.0003968253968253968 (* im im) 0.016666666666666666)
          (* im im)
          0.3333333333333333)
         (* im im)
         2.0)
        im))
      (* 0.5 re))
     (if (<= t_0 1e-7)
       (* (- (sin re)) im)
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -5e-306) {
		tmp = -(fma(fma(fma(0.0003968253968253968, (im * im), 0.016666666666666666), (im * im), 0.3333333333333333), (im * im), 2.0) * im) * (0.5 * re);
	} else if (t_0 <= 1e-7) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -5e-306)
		tmp = Float64(Float64(-Float64(fma(fma(fma(0.0003968253968253968, Float64(im * im), 0.016666666666666666), Float64(im * im), 0.3333333333333333), Float64(im * im), 2.0) * im)) * Float64(0.5 * re));
	elseif (t_0 <= 1e-7)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-306], N[((-N[(N[(N[(N[(0.0003968253968253968 * N[(im * im), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * im), $MachinePrecision]) * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.99999999999999998e-306

   1. Initial program 97.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.2

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

   10. Applied rewrites 65.2%

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

   if -4.99999999999999998e-306 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 9.9999999999999995e-8

   1. Initial program 32.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 9.9999999999999995e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 77.7%

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

Alternative 5: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \cdot \left(e^{-im} - e^{im}\right) \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\left(\sinh im \cdot -2\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= (* t_0 (- (exp (- im)) (exp im))) -5e-306)
     (* (* (sinh im) -2.0) (* 0.5 re))
     (*
      t_0
      (*
       (-
        (*
         (-
          (*
           (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
           im)
          0.3333333333333333)
         (* im im))
        2.0)
       im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((t_0 * (exp(-im) - exp(im))) <= -5e-306) {
		tmp = (sinh(im) * -2.0) * (0.5 * re);
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if ((t_0 * (exp(-im) - exp(im))) <= (-5d-306)) then
        tmp = (sinh(im) * (-2.0d0)) * (0.5d0 * re)
    else
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if ((t_0 * (Math.exp(-im) - Math.exp(im))) <= -5e-306) {
		tmp = (Math.sinh(im) * -2.0) * (0.5 * re);
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 * (math.exp(-im) - math.exp(im))) <= -5e-306:
		tmp = (math.sinh(im) * -2.0) * (0.5 * re)
	else:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im)) - exp(im))) <= -5e-306)
		tmp = Float64(Float64(sinh(im) * -2.0) * Float64(0.5 * re));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 * (exp(-im) - exp(im))) <= -5e-306)
		tmp = (sinh(im) * -2.0) * (0.5 * re);
	else
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-306], N[(N[(N[Sinh[im], $MachinePrecision] * -2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.99999999999999998e-306

   1. Initial program 97.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 76.8

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

   7. Applied rewrites 76.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 76.8

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

   9. Applied rewrites 76.8%

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

   if -4.99999999999999998e-306 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 56.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

Alternative 6: 88.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 76.3

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

   9. Applied rewrites 76.3%

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

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

   1. Initial program 56.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.0%

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

Alternative 7: 59.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 5e-15)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 5e-15) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 5e-15)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 5e-15], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 4.99999999999999999e-15

   1. Initial program 70.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 69.9

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 75.2%

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

   if 4.99999999999999999e-15 < (sin.f64 re)

   1. Initial program 48.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 25.3

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

   8. Applied rewrites 25.3%

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

Alternative 8: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 5e-15)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (* (- (* (* -0.3333333333333333 im) im) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 5e-15) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((-0.3333333333333333 * im) * im) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 5e-15)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(-0.3333333333333333 * im) * im) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 5e-15], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 4.99999999999999999e-15

   1. Initial program 70.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 69.9

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 75.2%

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

   if 4.99999999999999999e-15 < (sin.f64 re)

   1. Initial program 48.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 25.3

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

   13. Applied rewrites 25.3%

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

Alternative 9: 59.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, im \cdot im, 0.016666666666666666\right), im \cdot im, 0.3333333333333333\right), im \cdot im, 2\right) \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (-
     (*
      (fma
       (fma
        (fma 0.0003968253968253968 (* im im) 0.016666666666666666)
        (* im im)
        0.3333333333333333)
       (* im im)
       2.0)
      im))
    (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = -(fma(fma(fma(0.0003968253968253968, (im * im), 0.016666666666666666), (im * im), 0.3333333333333333), (im * im), 2.0) * im) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(-Float64(fma(fma(fma(0.0003968253968253968, Float64(im * im), 0.016666666666666666), Float64(im * im), 0.3333333333333333), Float64(im * im), 2.0) * im)) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(N[(0.0003968253968253968 * N[(im * im), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * im), $MachinePrecision]) * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -5.0000000000000001e-4

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 29.8

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

   8. Applied rewrites 29.8%

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

   if -5.0000000000000001e-4 < (sin.f64 re)

   1. Initial program 67.8%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 80.8

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 77.7

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

   10. Applied rewrites 77.7%

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

Alternative 10: 59.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, im \cdot im, 0.016666666666666666\right), im \cdot im, 0.3333333333333333\right), im \cdot im, 2\right) \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (- (* (* -0.3333333333333333 im) im) 2.0) im))
   (*
    (-
     (*
      (fma
       (fma
        (fma 0.0003968253968253968 (* im im) 0.016666666666666666)
        (* im im)
        0.3333333333333333)
       (* im im)
       2.0)
      im))
    (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((-0.3333333333333333 * im) * im) - 2.0) * im);
	} else {
		tmp = -(fma(fma(fma(0.0003968253968253968, (im * im), 0.016666666666666666), (im * im), 0.3333333333333333), (im * im), 2.0) * im) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(-0.3333333333333333 * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(-Float64(fma(fma(fma(0.0003968253968253968, Float64(im * im), 0.016666666666666666), Float64(im * im), 0.3333333333333333), Float64(im * im), 2.0) * im)) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(N[(0.0003968253968253968 * N[(im * im), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * im), $MachinePrecision]) * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -5.0000000000000001e-4

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 29.8

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

   8. Applied rewrites 29.8%

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

   10. Applied rewrites 29.8%

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

   if -5.0000000000000001e-4 < (sin.f64 re)

   1. Initial program 67.8%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 80.8

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 77.7

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

   10. Applied rewrites 77.7%

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

Alternative 11: 55.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-239}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 1e-239)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (- (* (* -0.3333333333333333 im) im) 2.0) im))
   (*
    (*
     (-
      (* (- (* (* -0.008333333333333333 im) im) 0.16666666666666666) (* im im))
      1.0)
     re)
    im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 1e-239) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((-0.3333333333333333 * im) * im) - 2.0) * im);
	} else {
		tmp = ((((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 1e-239)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(-0.3333333333333333 * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 1e-239], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.0000000000000001e-239

   1. Initial program 70.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

   10. Applied rewrites 62.8%

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

   if 1.0000000000000001e-239 < (sin.f64 re)

   1. Initial program 59.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (*
    (* (* (* re re) -0.08333333333333333) re)
    (* (- (* (* im im) -0.3333333333333333) 2.0) im))
   (*
    (* im re)
    (-
     (* (- (* (* -0.008333333333333333 im) im) 0.16666666666666666) (* im im))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = (((re * re) * -0.08333333333333333) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	} else {
		tmp = (im * re) * (((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * (im * im)) - 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (sin(re) <= (-0.0005d0)) then
        tmp = (((re * re) * (-0.08333333333333333d0)) * re) * ((((im * im) * (-0.3333333333333333d0)) - 2.0d0) * im)
    else
        tmp = (im * re) * ((((((-0.008333333333333333d0) * im) * im) - 0.16666666666666666d0) * (im * im)) - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sin(re) <= -0.0005) {
		tmp = (((re * re) * -0.08333333333333333) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	} else {
		tmp = (im * re) * (((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * (im * im)) - 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sin(re) <= -0.0005:
		tmp = (((re * re) * -0.08333333333333333) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im)
	else:
		tmp = (im * re) * (((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * (im * im)) - 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	else
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sin(re) <= -0.0005)
		tmp = (((re * re) * -0.08333333333333333) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	else
		tmp = (im * re) * (((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * (im * im)) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -5.0000000000000001e-4

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.7%

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

   if -5.0000000000000001e-4 < (sin.f64 re)

   1. Initial program 67.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.2%

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

Alternative 13: 55.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.16666666666666666 \cdot re\right), re, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (* (fma (* im (* 0.16666666666666666 re)) re (- im)) re)
   (*
    (* im re)
    (-
     (* (- (* (* -0.008333333333333333 im) im) 0.16666666666666666) (* im im))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = fma((im * (0.16666666666666666 * re)), re, -im) * re;
	} else {
		tmp = (im * re) * (((((-0.008333333333333333 * im) * im) - 0.16666666666666666) * (im * im)) - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(fma(Float64(im * Float64(0.16666666666666666 * re)), re, Float64(-im)) * re);
	else
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(im * N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -5.0000000000000001e-4

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   10. Applied rewrites 25.5%

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

   if -5.0000000000000001e-4 < (sin.f64 re)

   1. Initial program 67.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.2%

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

Alternative 14: 53.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.16666666666666666 \cdot re\right), re, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (* (fma (* im (* 0.16666666666666666 re)) re (- im)) re)
   (* (* 0.5 re) (* (- (* (* im im) -0.3333333333333333) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = fma((im * (0.16666666666666666 * re)), re, -im) * re;
	} else {
		tmp = (0.5 * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(fma(Float64(im * Float64(0.16666666666666666 * re)), re, Float64(-im)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(im * N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -5.0000000000000001e-4

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   10. Applied rewrites 25.5%

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

   if -5.0000000000000001e-4 < (sin.f64 re)

   1. Initial program 67.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

Alternative 15: 34.3% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 1e-6)
   (* (fma (* im (* 0.16666666666666666 re)) re (- im)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 1e-6) {
		tmp = fma((im * (0.16666666666666666 * re)), re, -im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 1e-6)
		tmp = Float64(fma(Float64(im * Float64(0.16666666666666666 * re)), re, Float64(-im)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 1e-6], N[(N[(N[(im * N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 9.99999999999999955e-7

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.7%

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

   10. Applied rewrites 45.7%

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

   if 9.99999999999999955e-7 < (sin.f64 re)

   1. Initial program 50.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.1%

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

Alternative 16: 34.3% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 1e-6)
   (* (fma 0.16666666666666666 (* (* re re) im) (- im)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 1e-6) {
		tmp = fma(0.16666666666666666, ((re * re) * im), -im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 1e-6)
		tmp = Float64(fma(0.16666666666666666, Float64(Float64(re * re) * im), Float64(-im)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 1e-6], N[(N[(0.16666666666666666 * N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 9.99999999999999955e-7

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.7%

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

   10. Applied rewrites 45.7%

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

   if 9.99999999999999955e-7 < (sin.f64 re)

   1. Initial program 50.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.1%

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

Alternative 17: 34.3% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 1e-6)
   (* (* im (fma (* 0.16666666666666666 re) re -1.0)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 1e-6) {
		tmp = (im * fma((0.16666666666666666 * re), re, -1.0)) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 1e-6)
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * re), re, -1.0)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 1e-6], N[(N[(im * N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 9.99999999999999955e-7

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.7%

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

   if 9.99999999999999955e-7 < (sin.f64 re)

   1. Initial program 50.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.1%

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

Alternative 18: 34.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (* (* (* (* im re) re) 0.16666666666666666) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (sin(re) <= (-0.0005d0)) then
        tmp = (((im * re) * re) * 0.16666666666666666d0) * re
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sin(re) <= -0.0005) {
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sin(re) <= -0.0005:
		tmp = (((im * re) * re) * 0.16666666666666666) * re
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(Float64(Float64(Float64(im * re) * re) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sin(re) <= -0.0005)
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -5.0000000000000001e-4

   1. Initial program 62.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.3%

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

   if -5.0000000000000001e-4 < (sin.f64 re)

   1. Initial program 67.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.3%

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

Alternative 19: 32.4% accurate, 39.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 52.0

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 37.8%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024326 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))