math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.9s

Alternatives: 20

Speedup: 1.5×

• 50.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Cosh[im], $MachinePrecision] * N[Sin[re], $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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. exp-0 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-0 N/A

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

   13. cosh-lowering-cosh.f64 N/A

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

   14. sin-lowering-sin.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* re (* re re)) (* (* (* im im) (* im im)) -0.006944444444444444))
     (if (<= t_0 1.0)
       (*
        (sin re)
        (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
       (* (cosh im) re)))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(Float64(im * im) * Float64(im * im)) * -0.006944444444444444));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 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 im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

   8. Simplified 66.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-lowering-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. distribute-lft-in N/A

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

   11. Simplified 25.5%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{3}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{144} \cdot {im}^{4}\right) \cdot {re}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       11. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{-1}{144}\right) \]

       12. pow-sqr N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       16. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{-1}{144}\right) \]

       17. *-lowering-*.f64 25.5

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.006944444444444444\right) \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 78.7%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 78.7

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

   9. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\cosh im} \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (* re (* re re)) (* (* (* im im) (* im im)) -0.006944444444444444))
     (if (<= t_1 1.0) (* t_0 (fma im im 2.0)) (* (cosh im) re)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(Float64(im * im) * Float64(im * im)) * -0.006944444444444444));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $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[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $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 im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

   8. Simplified 66.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-lowering-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. distribute-lft-in N/A

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

   11. Simplified 25.5%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{3}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{144} \cdot {im}^{4}\right) \cdot {re}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       11. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{-1}{144}\right) \]

       12. pow-sqr N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       16. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{-1}{144}\right) \]

       17. *-lowering-*.f64 25.5

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.006944444444444444\right) \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 78.7%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 78.7

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

   9. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\cosh im} \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* re (* re re)) (* (* (* im im) (* im im)) -0.006944444444444444))
     (if (<= t_0 1.0) (sin re) (* (cosh im) re)))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (Math.sin(re) * 0.5) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.cosh(im) * re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.sin(re) * 0.5) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444)
	elif t_0 <= 1.0:
		tmp = math.sin(re)
	else:
		tmp = math.cosh(im) * re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(Float64(im * im) * Float64(im * im)) * -0.006944444444444444));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = cosh(im) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[Cosh[im], $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 im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

   8. Simplified 66.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-lowering-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. distribute-lft-in N/A

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

   11. Simplified 25.5%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{3}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{144} \cdot {im}^{4}\right) \cdot {re}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       11. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{-1}{144}\right) \]

       12. pow-sqr N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       16. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{-1}{144}\right) \]

       17. *-lowering-*.f64 25.5

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.006944444444444444\right) \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 99.1

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

   5. Simplified 99.1%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 78.7%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 78.7

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

   9. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\cosh im} \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* re (* re re)) (* (* (* im im) (* im im)) -0.006944444444444444))
     (if (<= t_0 1.0)
       (sin re)
       (*
        re
        (fma
         (* im im)
         (fma
          im
          (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
          0.5)
         1.0))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = re * fma((im * im), fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(Float64(im * im) * Float64(im * im)) * -0.006944444444444444));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(re * fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

   8. Simplified 66.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-lowering-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. distribute-lft-in N/A

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

   11. Simplified 25.5%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{3}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{144} \cdot {im}^{4}\right) \cdot {re}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       11. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{-1}{144}\right) \]

       12. pow-sqr N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       16. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{-1}{144}\right) \]

       17. *-lowering-*.f64 25.5

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.006944444444444444\right) \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 99.1

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

   5. Simplified 99.1%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 78.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 74.0

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

   10. Simplified 74.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (fma im im 2.0) (* re (* (* re re) -0.08333333333333333)))
     (if (<= t_0 0.001)
       (fma re (* (* re re) -0.16666666666666666) re)
       (* re (* (* im im) (fma im (* im 0.041666666666666664) 0.5)))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * (re * ((re * re) * -0.08333333333333333));
	} else if (t_0 <= 0.001) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * ((im * im) * fma(im, (im * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * Float64(re * Float64(Float64(re * re) * -0.08333333333333333)));
	elseif (t_0 <= 0.001)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(Float64(im * im) * fma(im, Float64(im * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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. accelerator-lowering-fma.f64 46.5

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

   5. Simplified 46.5%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. +-commutative N/A

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

       8. unpow2 N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 24.6

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

   11. Simplified 24.6%

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

   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))) < 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 im around 0

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

      1. sin-lowering-sin.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 61.2

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

   8. Simplified 61.2%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 82.2%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 48.6%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{1}{24} \cdot re\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{re}{{im}^{2}} \cdot {im}^{4}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{re \cdot {im}^{4}}{{im}^{2}}} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       5. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot \frac{{im}^{4}}{{im}^{2}}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       7. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       8. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{{im}^{2}}{{im}^{2}}\right)}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       10. associate-*r/ N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       11. rgt-mult-inverse N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{1}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       12. *-rgt-identity N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       14. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(re \cdot \frac{1}{24}\right)} \cdot {im}^{4} \]

       16. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{re \cdot \left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       17. metadata-eval N/A

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

       18. pow-sqr N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

   11. Simplified 48.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* im im) (* re (fma -0.08333333333333333 (* re re) 0.5)))
     (if (<= t_0 0.001)
       (fma re (* (* re re) -0.16666666666666666) re)
       (* re (* (* im im) (fma im (* im 0.041666666666666664) 0.5)))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (im * im) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else if (t_0 <= 0.001) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * ((im * im) * fma(im, (im * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(im * im) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	elseif (t_0 <= 0.001)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(Float64(im * im) * fma(im, Float64(im * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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. accelerator-lowering-fma.f64 46.5

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

   5. Simplified 46.5%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 46.5

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

   11. Simplified 46.5%

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

   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))) < 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 im around 0

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

      1. sin-lowering-sin.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 61.2

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

   8. Simplified 61.2%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 82.2%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 48.6%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{1}{24} \cdot re\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{re}{{im}^{2}} \cdot {im}^{4}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{re \cdot {im}^{4}}{{im}^{2}}} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       5. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot \frac{{im}^{4}}{{im}^{2}}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       7. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       8. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{{im}^{2}}{{im}^{2}}\right)}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       10. associate-*r/ N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       11. rgt-mult-inverse N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{1}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       12. *-rgt-identity N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       14. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(re \cdot \frac{1}{24}\right)} \cdot {im}^{4} \]

       16. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{re \cdot \left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       17. metadata-eval N/A

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

       18. pow-sqr N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

   11. Simplified 48.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 95.5%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 78.7%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 78.7

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

   9. Applied egg-rr 78.7%

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

4. Final simplification 91.5%

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

Alternative 9: 58.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

   8. Simplified 63.7%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 60.6%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 57.0

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

   10. Simplified 57.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 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 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. accelerator-lowering-fma.f64 76.0

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

   5. Simplified 76.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.9

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

   8. Simplified 54.9%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 60.6%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 57.0

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

   10. Simplified 57.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.001)
   (* (fma -0.08333333333333333 (* re re) 0.5) (* re (fma im im 2.0)))
   (* re (* (* im im) (fma im (* im 0.041666666666666664) 0.5)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 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 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. accelerator-lowering-fma.f64 76.0

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

   5. Simplified 76.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.9

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

   8. Simplified 54.9%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 82.2%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 48.6%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{1}{24} \cdot re\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{re}{{im}^{2}} \cdot {im}^{4}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{re \cdot {im}^{4}}{{im}^{2}}} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       5. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot \frac{{im}^{4}}{{im}^{2}}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       7. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       8. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{{im}^{2}}{{im}^{2}}\right)}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       10. associate-*r/ N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       11. rgt-mult-inverse N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{1}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       12. *-rgt-identity N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       14. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(re \cdot \frac{1}{24}\right)} \cdot {im}^{4} \]

       16. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{re \cdot \left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       17. metadata-eval N/A

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

       18. pow-sqr N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

   11. Simplified 48.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.001)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* re (* (* im im) (fma im (* im 0.041666666666666664) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.001) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * ((im * im) * fma(im, (im * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.001)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(Float64(im * im) * fma(im, Float64(im * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

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

      1. sin-lowering-sin.f64 55.7

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

   5. Simplified 55.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 42.4

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

   8. Simplified 42.4%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 82.2%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 48.6%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{1}{24} \cdot re\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot {im}^{4} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4}} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{re}{{im}^{2}} \cdot {im}^{4}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{re \cdot {im}^{4}}{{im}^{2}}} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       5. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot \frac{{im}^{4}}{{im}^{2}}\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       7. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       8. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{{im}^{2}}{{im}^{2}}\right)}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       10. associate-*r/ N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       11. rgt-mult-inverse N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{1}\right)\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       12. *-rgt-identity N/A

           \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{{im}^{2}}\right) + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot re\right)} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       14. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} + \left(\frac{1}{24} \cdot re\right) \cdot {im}^{4} \]

       15. *-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(re \cdot \frac{1}{24}\right)} \cdot {im}^{4} \]

       16. associate-*l* N/A

           \[\leadsto \left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + \color{blue}{re \cdot \left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       17. metadata-eval N/A

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

       18. pow-sqr N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

   11. Simplified 48.8%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.001)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* re (* (* im im) (* im (* im 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.001) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * ((im * im) * (im * (im * 0.041666666666666664)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.001)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

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

      1. sin-lowering-sin.f64 55.7

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

   5. Simplified 55.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 42.4

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

   8. Simplified 42.4%

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

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 82.2%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 48.6%

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

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 48.8

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

   11. Simplified 48.8%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.001)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* im (* im (* 0.041666666666666664 (* re (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.001) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = im * (im * (0.041666666666666664 * (re * (im * im))));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.001)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(im * Float64(im * Float64(0.041666666666666664 * Float64(re * Float64(im * im)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

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

      1. sin-lowering-sin.f64 55.7

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

   5. Simplified 55.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. *-lowering-*.f64 42.4

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 42.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \frac{1}{12} \cdot \color{blue}{\left(im \cdot im\right)}, im\right), 2\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right) \cdot im}, im\right), 2\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot im\right)}, im\right), 2\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot im\right)}, im\right), 2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(im \cdot \frac{1}{12}\right)}, im\right), 2\right) \]

      14. *-lowering-*.f64 82.2

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(im \cdot 0.08333333333333333\right)}, im\right), 2\right) \]

   5. Simplified 82.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \left(im \cdot 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(2 + \color{blue}{\left(im \cdot im + \left(\frac{1}{12} \cdot {im}^{3}\right) \cdot im\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(2 + \left(\color{blue}{{im}^{2}} + \left(\frac{1}{12} \cdot {im}^{3}\right) \cdot im\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(\left(2 + {im}^{2}\right) + \left(\frac{1}{12} \cdot {im}^{3}\right) \cdot im\right)} \]

      7. unpow3 N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\left(2 + {im}^{2}\right) + \left(\frac{1}{12} \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)}\right) \cdot im\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\left(2 + {im}^{2}\right) + \left(\frac{1}{12} \cdot \left(\color{blue}{{im}^{2}} \cdot im\right)\right) \cdot im\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\left(2 + {im}^{2}\right) + \color{blue}{\left(\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot im\right)} \cdot im\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\left(2 + {im}^{2}\right) + \color{blue}{\left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \left(im \cdot im\right)}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\left(2 + {im}^{2}\right) + \left(\frac{1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      12. associate-+r+ N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + \left(\frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2}\right)\right)} \]

      13. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(2 + \left(\color{blue}{1 \cdot {im}^{2}} + \left(\frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2}\right)\right) \]

      14. distribute-rgt-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(2 + \color{blue}{{im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)}\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, 1 + \frac{1}{12} \cdot {im}^{2}, 2\right)} \]

   8. Simplified 48.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), 2\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot re} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot re \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \cdot re \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot re \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right) \]

       8. *-commutative N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)} \cdot re\right) \]

       9. associate-*r* N/A

          \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot re\right)\right)} \]

       10. associate-*l* N/A

           \[\leadsto \color{blue}{im \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot re\right)\right)\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{im \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot re\right)\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot re\right)\right)\right)} \]

       13. associate-*r* N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\left({im}^{2} \cdot \frac{1}{24}\right) \cdot re\right)}\right) \]

       14. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)} \cdot re\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)}\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)}\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot re\right)}\right)\right) \]

       18. unpow2 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot re\right)\right)\right) \]

       19. *-lowering-*.f64 42.9

           \[\leadsto im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot re\right)\right)\right) \]

   11. Simplified 42.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.001)
   (fma re (* (* re re) -0.16666666666666666) re)
   (fma (* 0.5 (* im im)) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.001) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = fma((0.5 * (im * im)), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.001)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = fma(Float64(0.5 * Float64(im * im)), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 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 im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 55.7

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 55.7%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. *-lowering-*.f64 42.4

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 42.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 1e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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. accelerator-lowering-fma.f64 62.9

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Simplified 62.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {im}^{2}\right) \cdot re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right)\right) \cdot re} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot {im}^{2}\right)} \cdot re \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} + \frac{1}{2} \cdot {im}^{2}\right) \cdot re \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \cdot re \]

      6. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re + re} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot {im}^{2}, re, re\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot {im}^{2}}, re, re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re, re\right) \]

      10. *-lowering-*.f64 37.7

          \[\leadsto \mathsf{fma}\left(0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re, re\right) \]

   8. Simplified 37.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)\\ \mathbf{elif}\;\sin re \leq 5 \cdot 10^{-14}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.01)
   (* (* re (* re re)) (* (* (* im im) (* im im)) -0.006944444444444444))
   (if (<= (sin re) 5e-14)
     (* re (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
     (*
      re
      (*
       (fma im im 2.0)
       (fma
        (* re re)
        (fma (* re re) 0.004166666666666667 -0.08333333333333333)
        0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = (re * (re * re)) * (((im * im) * (im * im)) * -0.006944444444444444);
	} else if (sin(re) <= 5e-14) {
		tmp = re * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = re * (fma(im, im, 2.0) * fma((re * re), fma((re * re), 0.004166666666666667, -0.08333333333333333), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(Float64(im * im) * Float64(im * im)) * -0.006944444444444444));
	elseif (sin(re) <= 5e-14)
		tmp = Float64(re * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(re * Float64(fma(im, im, 2.0) * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[re], $MachinePrecision], 5e-14], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \frac{1}{12} \cdot \color{blue}{\left(im \cdot im\right)}, im\right), 2\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(\frac{1}{12} \cdot im\right) \cdot im}, im\right), 2\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot im\right)}, im\right), 2\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot im\right)}, im\right), 2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(im \cdot \frac{1}{12}\right)}, im\right), 2\right) \]

      14. *-lowering-*.f64 87.7

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(im \cdot 0.08333333333333333\right)}, im\right), 2\right) \]

   5. Simplified 87.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \left(im \cdot 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) + \frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right) + re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(re \cdot \frac{1}{2}\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)} + re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) + re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) + re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) + \color{blue}{\left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)} \]

      7. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) \cdot \left(\frac{1}{2} \cdot re + re \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) \cdot \left(\color{blue}{re \cdot \frac{1}{2}} + re \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

   8. Simplified 28.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({re}^{3} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right) \cdot \frac{-1}{12}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{12}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

       4. metadata-eval N/A

          \[\leadsto {re}^{3} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{12}\right)\right)} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right) \]

       5. distribute-lft-neg-in N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{12}\right)\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right)} \]

       13. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\frac{-1}{12}} \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)\right) \]

       14. +-commutative N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)}\right) \]

       15. distribute-lft-in N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) + \frac{-1}{12} \cdot 2\right)} \]

   11. Simplified 27.2%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.08333333333333333, \left(im \cdot im\right) \cdot 0.08333333333333333, -0.08333333333333333\right), -0.16666666666666666\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({im}^{4} \cdot {re}^{3}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{144} \cdot {im}^{4}\right) \cdot {re}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{144} \cdot {im}^{4}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left({im}^{4} \cdot \frac{-1}{144}\right)} \]

       11. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{-1}{144}\right) \]

       12. pow-sqr N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{-1}{144}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \frac{-1}{144}\right) \]

       16. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \frac{-1}{144}\right) \]

       17. *-lowering-*.f64 26.6

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot -0.006944444444444444\right) \]

   14. Simplified 26.6%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.006944444444444444\right)} \]

   if -0.0100000000000000002 < (sin.f64 re) < 5.0000000000000002e-14

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 87.8%

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if 5.0000000000000002e-14 < (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. accelerator-lowering-fma.f64 67.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Simplified 67.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{12} \cdot \left(2 + {im}^{2}\right) + \frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{12} \cdot \left(2 + {im}^{2}\right) + \frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(2 + {im}^{2}\right) \cdot \frac{1}{2}} + {re}^{2} \cdot \left(\frac{-1}{12} \cdot \left(2 + {im}^{2}\right) + \frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{12} \cdot \left(2 + {im}^{2}\right)\right) + {re}^{2} \cdot \left(\frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{\left({re}^{2} \cdot \frac{-1}{12}\right) \cdot \left(2 + {im}^{2}\right)} + {re}^{2} \cdot \left(\frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2}\right)} \cdot \left(2 + {im}^{2}\right) + {re}^{2} \cdot \left(\frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right) + {re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{240} \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right)\right)}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right) + \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2}\right)\right) \cdot \left(2 + {im}^{2}\right)}\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2}\right)\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(2 + {im}^{2}\right) \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2}\right) + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(2 + {im}^{2}\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2}\right) + \color{blue}{{re}^{2} \cdot \frac{-1}{12}}\right)\right) \]

      11. distribute-lft-in N/A

          \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(2 + {im}^{2}\right) \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} + \frac{-1}{12}\right)\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto re \cdot \left(\left(2 + {im}^{2}\right) \cdot \frac{1}{2} + \left(2 + {im}^{2}\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}\right)\right)\right) \]

   8. Simplified 33.4%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 56.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.01)
   (* (fma im im 2.0) (* re (* (* re re) -0.08333333333333333)))
   (* re (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * (re * ((re * re) * -0.08333333333333333));
	} else {
		tmp = re * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * Float64(re * Float64(Float64(re * re) * -0.08333333333333333)));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. 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. accelerator-lowering-fma.f64 74.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Simplified 74.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right) + \frac{1}{2} \cdot \left(2 + {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2}\right)\right)\right) \cdot re + \left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right)\right) \cdot re} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(2 + {im}^{2}\right)\right)} \cdot re + \left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right)\right) \cdot re \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(\left(2 + {im}^{2}\right) \cdot re\right)} + \left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right)\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot \left(2 + {im}^{2}\right)\right)} + \left(\frac{1}{2} \cdot \left(2 + {im}^{2}\right)\right) \cdot re \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(2 + {im}^{2}\right) \cdot re\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(2 + {im}^{2}\right)\right)} \]

      7. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(2 + {im}^{2}\right)\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \left(re \cdot \left(2 + {im}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(2 + {im}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(2 + {im}^{2}\right)\right)} \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({im}^{2} + 2\right)}\right) \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{im \cdot im} + 2\right)\right) \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{12}, {re}^{2}, \frac{1}{2}\right)} \]

      16. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{12}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \]

      17. *-lowering-*.f64 27.3

          \[\leadsto \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \mathsf{fma}\left(-0.08333333333333333, \color{blue}{re \cdot re}, 0.5\right) \]

   8. Simplified 27.3%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({re}^{3} \cdot \left(2 + {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{12} \cdot {re}^{3}\right) \cdot \left(2 + {im}^{2}\right)} \]

       2. unpow3 N/A

          \[\leadsto \left(\frac{-1}{12} \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot re\right)}\right) \cdot \left(2 + {im}^{2}\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{12} \cdot \left(\color{blue}{{re}^{2}} \cdot re\right)\right) \cdot \left(2 + {im}^{2}\right) \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(2 + {im}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 + {im}^{2}\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \]

       7. +-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} + 2\right)} \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{im \cdot im} + 2\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(\frac{-1}{12} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

       14. *-lowering-*.f64 26.0

           \[\leadsto \mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(-0.08333333333333333 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(-0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 71.5%

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.01) (* (* re (* re re)) -0.16666666666666666) re))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = (re * (re * re)) * -0.16666666666666666;
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (sin(re) <= (-0.01d0)) then
        tmp = (re * (re * re)) * (-0.16666666666666666d0)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sin(re) <= -0.01) {
		tmp = (re * (re * re)) * -0.16666666666666666;
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sin(re) <= -0.01:
		tmp = (re * (re * re)) * -0.16666666666666666
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(re * Float64(re * re)) * -0.16666666666666666);
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sin(re) <= -0.01)
		tmp = (re * (re * re)) * -0.16666666666666666;
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.9

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 48.9%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. *-lowering-*.f64 19.9

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 19.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(re \cdot {re}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       6. *-lowering-*.f64 19.8

          \[\leadsto -0.16666666666666666 \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   11. Simplified 19.8%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)} \]

   if -0.0100000000000000002 < (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

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 45.2

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 45.2%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 33.6%

      \[\leadsto \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma re (* (* re re) -0.16666666666666666) re))

C

double code(double re, double im) {
	return fma(re, ((re * re) * -0.16666666666666666), re);
}

Julia

function code(re, im)
	return fma(re, Float64(Float64(re * re) * -0.16666666666666666), re)
end

Wolfram

code[re_, im_] := N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 46.3

      \[\leadsto \color{blue}{\sin re} \]

5. Simplified 46.3%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

   8. *-lowering-*.f64 33.6

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

8. Simplified 33.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]
9. Add Preprocessing

Alternative 20: 27.6% accurate, 317.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re
end function

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 46.3

      \[\leadsto \color{blue}{\sin re} \]

5. Simplified 46.3%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 24.3%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))