math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 14

Speedup: 1.5×

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

Specification

Math

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

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, 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. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. cosh-0 N/A

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

   18. exp-0 N/A

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

   19. lower-*.f64 N/A

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

   20. exp-0 N/A

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

   21. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 86.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double t_0 = 2.0 * cosh(im);
	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 * (fma(-0.08333333333333333, (re * re), 0.5) * re);
	} else if (t_2 <= 1.0) {
		tmp = t_1 * fma(im, im, 2.0);
	} else {
		tmp = t_0 * (re * 0.5);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $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[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $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 * 0.5), $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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

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

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.5

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

   10. Applied rewrites 74.5%

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

4. Final simplification 87.3%

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

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       2.0
       (fma
        (fma
         (fma 0.001388888888888889 (* im im) 0.041666666666666664)
         (* im im)
         0.5)
        (* im im)
        1.0))
      (* (fma -0.08333333333333333 (* re re) 0.5) re))
     (if (<= t_1 1.0)
       (* t_0 (fma im im 2.0))
       (* (* 2.0 (cosh im)) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (2.0 * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0)) * (fma(-0.08333333333333333, (re * re), 0.5) * re);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = (2.0 * cosh(im)) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0)) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(2.0 * cosh(im)) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(2.0 * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      4. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      7. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      9. unpow2 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      11. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 67.0

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   10. Applied rewrites 67.0%

       \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

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

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.5

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

   10. Applied rewrites 74.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       2.0
       (fma
        (fma
         (fma 0.001388888888888889 (* im im) 0.041666666666666664)
         (* im im)
         0.5)
        (* im im)
        1.0))
      (* (fma -0.08333333333333333 (* re re) 0.5) re))
     (if (<= t_0 1.0) (sin re) (* (* 2.0 (cosh im)) (* re 0.5))))))

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 = (2.0 * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0)) * (fma(-0.08333333333333333, (re * re), 0.5) * re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (2.0 * cosh(im)) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0)) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(2.0 * cosh(im)) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(2.0 * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      4. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      7. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      9. unpow2 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      11. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 67.0

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   10. Applied rewrites 67.0%

       \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 61.7

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

   7. Applied rewrites 61.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-sin.f64 98.4

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

   10. Applied rewrites 98.4%

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

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.5

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

   10. Applied rewrites 74.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       2.0
       (fma
        (fma
         (fma 0.001388888888888889 (* im im) 0.041666666666666664)
         (* im im)
         0.5)
        (* im im)
        1.0))
      (* (fma -0.08333333333333333 (* re re) 0.5) re))
     (if (<= t_0 1.0)
       (sin re)
       (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (2.0 * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0)) * (fma(-0.08333333333333333, (re * re), 0.5) * re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0)) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(2.0 * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      4. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      7. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      9. unpow2 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      11. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 67.0

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   10. Applied rewrites 67.0%

       \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 61.7

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

   7. Applied rewrites 61.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-sin.f64 98.4

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

   10. Applied rewrites 98.4%

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

   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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 79.6

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

   5. Applied rewrites 79.6%

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

   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 68.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 0.001:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot re, re, \left(\left(\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 0.001)
   (*
    (*
     2.0
     (fma
      (fma
       (fma 0.001388888888888889 (* im im) 0.041666666666666664)
       (* im im)
       0.5)
      (* im im)
      1.0))
    (* (fma -0.08333333333333333 (* re re) 0.5) re))
   (*
    (fma
     (-
      (fma
       (* 0.008333333333333333 re)
       re
       (*
        (*
         (* (- (* 0.008333333333333333 (* re re)) 0.16666666666666666) 0.5)
         im)
        im))
      0.16666666666666666)
     (* re re)
     (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 0.001) {
		tmp = (2.0 * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0)) * (fma(-0.08333333333333333, (re * re), 0.5) * re);
	} else {
		tmp = fma((fma((0.008333333333333333 * re), re, (((((0.008333333333333333 * (re * re)) - 0.16666666666666666) * 0.5) * im) * im)) - 0.16666666666666666), (re * re), fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 0.001)
		tmp = Float64(Float64(2.0 * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0)) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.008333333333333333 * re), re, Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666) * 0.5) * im) * im)) - 0.16666666666666666), Float64(re * re), fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 0.001], N[(N[(2.0 * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * re), $MachinePrecision] * re + N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 78.3

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      4. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      7. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      9. unpow2 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      11. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 74.2

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   if 1e-3 < (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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 89.7

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 20.1%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 20.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot re, re, \left(\left(\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) - 0.16666666666666666, re \cdot re, \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 5e-15)
   (*
    (*
     2.0
     (fma
      (fma
       (fma 0.001388888888888889 (* im im) 0.041666666666666664)
       (* im im)
       0.5)
      (* im im)
      1.0))
    (* (fma -0.08333333333333333 (* re re) 0.5) re))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 5e-15) {
		tmp = (2.0 * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0)) * (fma(-0.08333333333333333, (re * re), 0.5) * re);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 5e-15)
		tmp = Float64(Float64(2.0 * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0)) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 5e-15], N[(N[(2.0 * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{0 - im}} + e^{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(e^{0 - im} + \color{blue}{e^{im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      15. sub0-neg N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      4. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      7. +-commutative N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      9. unpow2 N/A

         \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      11. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 73.8

          \[\leadsto \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   10. Applied rewrites 73.8%

       \[\leadsto \left(2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \]

   if 4.99999999999999999e-15 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

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

Alternative 8: 55.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if -5.0000000000000001e-4 < (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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 88.0

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

   5. Applied rewrites 88.0%

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

   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 75.0%

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

Alternative 9: 55.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.0005)
   (* (* (* (* re re) -0.08333333333333333) re) (fma im im 2.0))
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.0005) {
		tmp = (((re * re) * -0.08333333333333333) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.0005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.0005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.2%

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

   if -5.0000000000000001e-4 < (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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 88.0

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

   5. Applied rewrites 88.0%

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

   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 75.0%

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

Alternative 10: 48.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.2%

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

   if -5.0000000000000001e-4 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 63.7

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

   8. Applied rewrites 63.7%

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

Alternative 11: 47.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 16.8

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

   8. Applied rewrites 16.8%

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

   10. Applied rewrites 16.8%

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

   if -5.0000000000000001e-4 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 63.7

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

   8. Applied rewrites 63.7%

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

Alternative 12: 47.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 16.8

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

   8. Applied rewrites 16.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 16.8%

       \[\leadsto \left(\left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right) \cdot re\right) \cdot 2 \]

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 16.8%

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

   if -5.0000000000000001e-4 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 63.7

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

   8. Applied rewrites 63.7%

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

Alternative 13: 48.0% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
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 77.4

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

5. Applied rewrites 77.4%

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

6. Taylor expanded in re around 0

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

   1. lower-*.f64 52.6

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

8. Applied rewrites 52.6%

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

Alternative 14: 25.9% accurate, 28.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 51.3%

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

6. Taylor expanded in re around 0

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

   1. lower-*.f64 31.8

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

8. Applied rewrites 31.8%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))