math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 20

Speedup: 1.5×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 100.0

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

   14. lift--.f64 N/A

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

   15. sub0-neg N/A

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

   16. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \sin re, e^{im}, 0.5 \cdot \left(\sin re \cdot e^{-im}\right)\right)} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp im) (exp (- im))))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       im
       (fma
        (fma (* im im) 0.002777777777777778 0.08333333333333333)
        (* im (* im im))
        im)
       2.0)
      (* re (fma re (* re -0.08333333333333333) 0.5)))
     (if (<= t_0 1.0)
       (*
        (sin re)
        (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
       (*
        (cosh im)
        (fma
         (fma re (* re 0.008333333333333333) -0.16666666666666666)
         (* re (* re re))
         re))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(im) + exp(-im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, fma(fma((im * im), 0.002777777777777778, 0.08333333333333333), (im * (im * im)), im), 2.0) * (re * fma(re, (re * -0.08333333333333333), 0.5));
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im) * fma(fma(re, (re * 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0) * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * fma(fma(re, Float64(re * 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * N[(N[(re * N[(re * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 75.4

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

   7. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 75.4

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

   9. Applied rewrites 75.4%

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

   10. Taylor expanded in re around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. unpow3 N/A

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

       8. lower-fma.f64 N/A

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

       9. sub-neg N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. cube-mult N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 73.7

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

   12. Applied rewrites 73.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ t_1 := 0.5 \cdot \sin re\\ t_2 := t\_1 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          im
          (fma
           (fma (* im im) 0.002777777777777778 0.08333333333333333)
           (* im (* im im))
           im)
          2.0))
        (t_1 (* 0.5 (sin re)))
        (t_2 (* t_1 (+ (exp im) (exp (- im))))))
   (if (<= t_2 (- INFINITY))
     (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5)))
     (if (<= t_2 1.0)
       (* t_1 (fma im im 2.0))
       (*
        t_0
        (*
         re
         (fma
          (* re re)
          (fma re (* re 0.004166666666666667) -0.08333333333333333)
          0.5)))))))

C

double code(double re, double im) {
	double t_0 = fma(im, fma(fma((im * im), 0.002777777777777778, 0.08333333333333333), (im * (im * im)), im), 2.0);
	double t_1 = 0.5 * sin(re);
	double t_2 = t_1 * (exp(im) + exp(-im));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	} else if (t_2 <= 1.0) {
		tmp = t_1 * fma(im, im, 2.0);
	} else {
		tmp = t_0 * (re * fma((re * re), fma(re, (re * 0.004166666666666667), -0.08333333333333333), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0)
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(t_1 * Float64(exp(im) + exp(Float64(-im))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	elseif (t_2 <= 1.0)
		tmp = Float64(t_1 * fma(im, im, 2.0));
	else
		tmp = Float64(t_0 * Float64(re * fma(Float64(re * re), fma(re, Float64(re * 0.004166666666666667), -0.08333333333333333), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(t$95$1 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * 0.004166666666666667), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      8. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{240} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{240}\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{240}\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      12. lower-*.f64 63.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.004166666666666667}, -0.08333333333333333\right), 0.5\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

   8. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      8. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{240} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{240}\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{240}\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      12. lower-*.f64 63.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.004166666666666667}, -0.08333333333333333\right), 0.5\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

   8. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= (* t_0 (+ (exp im) (exp (- im)))) 1.0)
     (*
      t_0
      (fma
       im
       (fma
        (fma (* im im) 0.002777777777777778 0.08333333333333333)
        (* im (* im im))
        im)
       2.0))
     (*
      (cosh im)
      (fma
       (fma re (* re 0.008333333333333333) -0.16666666666666666)
       (* re (* re re))
       re)))))

C

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

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(im) + exp(Float64(-im)))) <= 1.0)
		tmp = Float64(t_0 * fma(im, fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0));
	else
		tmp = Float64(cosh(im) * fma(fma(re, Float64(re * 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	end
	return 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], 1.0], N[(t$95$0 * N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * N[(N[(re * N[(re * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 97.6%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 75.4

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

   7. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 75.4

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

   9. Applied rewrites 75.4%

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

   10. Taylor expanded in re around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. unpow3 N/A

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

       8. lower-fma.f64 N/A

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

       9. sub-neg N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. cube-mult N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 73.7

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

   12. Applied rewrites 73.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.02)
   (*
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0)
    (fma re (* -0.16666666666666666 (* re re)) re))
   (*
    (fma
     im
     (fma
      (fma (* im im) 0.002777777777777778 0.08333333333333333)
      (* im (* im im))
      im)
     2.0)
    (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.02) {
		tmp = fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0) * fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = fma(im, fma(fma((im * im), 0.002777777777777778, 0.08333333333333333), (im * (im * im)), im), 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.02)
		tmp = Float64(fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0) * fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re));
	else
		tmp = Float64(fma(im, fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.7%

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

   if 0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, t\_0, 1\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot t\_0\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma im (* im 0.041666666666666664) 0.5)))
   (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.02)
     (* (fma (* im im) t_0 1.0) (fma re (* -0.16666666666666666 (* re re)) re))
     (fma (* im (* im t_0)) re re))))

C

double code(double re, double im) {
	double t_0 = fma(im, (im * 0.041666666666666664), 0.5);
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.02) {
		tmp = fma((im * im), t_0, 1.0) * fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = fma((im * (im * t_0)), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, Float64(im * 0.041666666666666664), 0.5)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.02)
		tmp = Float64(fma(Float64(im * im), t_0, 1.0) * fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re));
	else
		tmp = fma(Float64(im * Float64(im * t_0)), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(im * im), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.7%

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

   if 0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.4%

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

   10. Applied rewrites 39.6%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, t\_0, 1\right) \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot t\_0\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma im (* im 0.041666666666666664) 0.5)))
   (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) -0.005)
     (* (fma (* im im) t_0 1.0) (* re (* -0.16666666666666666 (* re re))))
     (fma (* im (* im t_0)) re re))))

C

double code(double re, double im) {
	double t_0 = fma(im, (im * 0.041666666666666664), 0.5);
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= -0.005) {
		tmp = fma((im * im), t_0, 1.0) * (re * (-0.16666666666666666 * (re * re)));
	} else {
		tmp = fma((im * (im * t_0)), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, Float64(im * 0.041666666666666664), 0.5)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= -0.005)
		tmp = Float64(fma(Float64(im * im), t_0, 1.0) * Float64(re * Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = fma(Float64(im * Float64(im * t_0)), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(im * im), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {re}^{3}\right) \cdot \mathsf{fma}\left(im \cdot \color{blue}{im}, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.6%

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

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 61.9%

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

   10. Applied rewrites 63.7%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.02)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (* re (* im (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.02) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.02)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.0%

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

   if 0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{24} \cdot \left({im}^{4} \cdot \color{blue}{re}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 36.8%

       \[\leadsto \left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot re\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 39.9%

       \[\leadsto \left(im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.041666666666666664\right) \cdot \left(im \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.02)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (* (* im 0.041666666666666664) (* im (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.02) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = (im * 0.041666666666666664) * (im * (re * (im * im)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.02)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = Float64(Float64(im * 0.041666666666666664) * Float64(im * Float64(re * Float64(im * 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], 0.02], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(im * 0.041666666666666664), $MachinePrecision] * N[(im * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.0%

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

   if 0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{24} \cdot \left({im}^{4} \cdot \color{blue}{re}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 36.8%

       \[\leadsto \left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot re\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 36.8%

       \[\leadsto \left(im \cdot 0.041666666666666664\right) \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.041666666666666664\right) \cdot \left(im \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub0-neg N/A

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

   13. cosh-undef N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

   16. exp-0 N/A

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

   17. lower-*.f64 N/A

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

   18. exp-0 N/A

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

   19. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \sin re \cdot \cosh im \]
6. Add Preprocessing

Alternative 12: 59.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ \mathbf{if}\;\sin re \leq 10^{-167}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          im
          (fma
           (fma (* im im) 0.002777777777777778 0.08333333333333333)
           (* im (* im im))
           im)
          2.0)))
   (if (<= (sin re) 1e-167)
     (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5)))
     (*
      t_0
      (*
       re
       (fma
        (* re re)
        (fma re (* re 0.004166666666666667) -0.08333333333333333)
        0.5))))))

C

double code(double re, double im) {
	double t_0 = fma(im, fma(fma((im * im), 0.002777777777777778, 0.08333333333333333), (im * (im * im)), im), 2.0);
	double tmp;
	if (sin(re) <= 1e-167) {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	} else {
		tmp = t_0 * (re * fma((re * re), fma(re, (re * 0.004166666666666667), -0.08333333333333333), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0)
	tmp = 0.0
	if (sin(re) <= 1e-167)
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	else
		tmp = Float64(t_0 * Float64(re * fma(Float64(re * re), fma(re, Float64(re * 0.004166666666666667), -0.08333333333333333), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[Sin[re], $MachinePrecision], 1e-167], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * 0.004166666666666667), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1e-167

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if 1e-167 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      8. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{240} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{240}\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{240}\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

      12. lower-*.f64 41.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.004166666666666667}, -0.08333333333333333\right), 0.5\right)\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ \mathbf{if}\;\sin re \leq 0.02:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          im
          (fma
           (fma (* im im) 0.002777777777777778 0.08333333333333333)
           (* im (* im im))
           im)
          2.0)))
   (if (<= (sin re) 0.02)
     (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5)))
     (* t_0 (* 0.5 re)))))

C

double code(double re, double im) {
	double t_0 = fma(im, fma(fma((im * im), 0.002777777777777778, 0.08333333333333333), (im * (im * im)), im), 2.0);
	double tmp;
	if (sin(re) <= 0.02) {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	} else {
		tmp = t_0 * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0)
	tmp = 0.0
	if (sin(re) <= 0.02)
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	else
		tmp = Float64(t_0 * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[Sin[re], $MachinePrecision], 0.02], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 74.3

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

   8. Applied rewrites 74.3%

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

   if 0.0200000000000000004 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 22.9

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

   8. Applied rewrites 22.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.005)
   (* (fma re (* -0.16666666666666666 (* re re)) re) (fma 0.5 (* im im) 1.0))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re) * fma(0.5, (im * im), 1.0);
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re) * fma(0.5, Float64(im * im), 1.0));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 29.5

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 27.9

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

   10. Applied rewrites 27.9%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 67.7%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.005)
   (* (fma re (* -0.16666666666666666 (* re re)) re) (* 0.5 (* im im)))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re) * (0.5 * (im * im));
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re) * Float64(0.5 * Float64(im * im)));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 29.5

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 27.9

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

   10. Applied rewrites 27.9%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 26.6%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 67.7%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.02 \cdot 10^{+14}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.02e+14)
   (* (cosh im) (fma re (* -0.16666666666666666 (* re re)) re))
   (* (sin re) (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.02e+14) {
		tmp = cosh(im) * fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = sin(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.02e+14)
		tmp = Float64(cosh(im) * fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re));
	else
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.02e+14], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.02e14

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.3

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

   7. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 78.3

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

   9. Applied rewrites 78.3%

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

   if 1.02e14 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 95.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.02 \cdot 10^{+14}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.005)
   (* re (* re (* re -0.16666666666666666)))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.3%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 20.3%

       \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 67.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.005)
   (* re (* re (* re -0.16666666666666666)))
   (fma 0.5 (* re (* im im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma(0.5, (re * (im * im)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(0.5, Float64(re * Float64(im * im)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.3%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 20.3%

       \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 57.8%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma re (* -0.16666666666666666 (* re re)) re))

C

double code(double re, double im) {
	return fma(re, (-0.16666666666666666 * (re * re)), re);
}

Julia

function code(re, im)
	return fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re)
end

Wolfram

code[re_, im_] := N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-sin.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 33.5%

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

9. Final simplification 33.5%

   \[\leadsto \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \]
10. Add Preprocessing

Alternative 20: 10.9% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (* re (* re -0.16666666666666666))))

C

double code(double re, double im) {
	return re * (re * (re * -0.16666666666666666));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (re * (re * (-0.16666666666666666d0)))
end function

Java

public static double code(double re, double im) {
	return re * (re * (re * -0.16666666666666666));
}

Python

def code(re, im):
	return re * (re * (re * -0.16666666666666666))

Julia

function code(re, im)
	return Float64(re * Float64(re * Float64(re * -0.16666666666666666)))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (re * (re * -0.16666666666666666));
end

Wolfram

code[re_, im_] := N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-sin.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 33.5%

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

9. Taylor expanded in re around inf

   \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 10.1%

    \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
12. Step-by-step derivation

13. Applied rewrites 10.1%

    \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

14. Final simplification 10.1%

    \[\leadsto re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))