math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 21.4s

Alternatives: 23

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       (* (sin im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
       (if (<= t_0 4e-96)
         t_1
         (if (<= t_0 1.0)
           (*
            (sin im)
            (fma
             (* (* re re) -0.25)
             (/ 1.0 (fma re 0.16666666666666666 -0.5))
             (+ re 1.0)))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 4e-96) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im) * fma(((re * re) * -0.25), (1.0 / fma(re, 0.16666666666666666, -0.5)), (re + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 4e-96)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) * fma(Float64(Float64(re * re) * -0.25), Float64(1.0 / fma(re, 0.16666666666666666, -0.5)), Float64(re + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-96], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(1.0 / N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 97.4

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

   5. Simplified 97.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   if 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. flip-+ N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

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

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)} \cdot \sin im \]

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]

      4. *-lowering-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right) \cdot \sin im \]

   10. Simplified 99.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right) \cdot -0.25}, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right) \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       (* (sin im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
       (if (<= t_0 4e-96)
         t_1
         (if (<= t_0 1.0)
           (*
            (sin im)
            (fma (fma re 0.16666666666666666 0.5) (* re re) (+ re 1.0)))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 4e-96) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im) * fma(fma(re, 0.16666666666666666, 0.5), (re * re), (re + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 4e-96)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), Float64(re + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-96], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 97.4

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

   5. Simplified 97.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   if 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. +-lowering-+.f64 99.8

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1
         (*
          (sin im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
        (t_2 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       t_1
       (if (<= t_0 4e-96) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 4e-96) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 4e-96)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 4e-96], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 98.7

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

   5. Simplified 98.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (sin im) (fma re (fma re 0.5 1.0) 1.0)))
        (t_2 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       t_1
       (if (<= t_0 4e-96) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 4e-96) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 4e-96)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 4e-96], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 98.4

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

   5. Simplified 98.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \sin im \cdot \left(re + 1\right)\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (sin im) (+ re 1.0)))
        (t_2 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       t_1
       (if (<= t_0 4e-96) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = sin(im) * (re + 1.0);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 4e-96) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(sin(im) * Float64(re + 1.0))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 4e-96)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 4e-96], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 2e-34) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 2e-34) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 2e-34)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-34], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 1.99999999999999986e-34 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 96.9

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

   5. Simplified 96.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999986e-34 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 96.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma re 0.16666666666666666 0.5) (* re re))
      (* -0.16666666666666666 (* im (* im im))))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 0.0)
         (*
          im
          (* im (* im (fma re -0.16666666666666666 -0.16666666666666666))))
         (if (<= t_0 1.0)
           (sin im)
           (*
            im
            (/
             (* (* re re) (fma (* re re) 0.027777777777777776 -0.25))
             (fma re 0.16666666666666666 -0.5)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(re, 0.16666666666666666, 0.5) * (re * re)) * (-0.16666666666666666 * (im * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = im * (im * (im * fma(re, -0.16666666666666666, -0.16666666666666666)));
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = im * (((re * re) * fma((re * re), 0.027777777777777776, -0.25)) / fma(re, 0.16666666666666666, -0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) * Float64(-0.16666666666666666 * Float64(im * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666))));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(Float64(re * re) * fma(Float64(re * re), 0.027777777777777776, -0.25)) / fma(re, 0.16666666666666666, -0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(im * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision]), $MachinePrecision] / N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 67.2

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

   8. Simplified 67.2%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft1-in N/A

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

       11. +-commutative N/A

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

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

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

   11. Simplified 53.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

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

       18. unpow3 N/A

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

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

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

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

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 15.8

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

   14. Simplified 15.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 96.8

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

   5. Simplified 96.8%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 38.0

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

   5. Simplified 38.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 37.8

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

   8. Simplified 37.8%

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

   9. Taylor expanded in im around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. sub-neg N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. accelerator-lowering-fma.f64 20.3

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

   11. Simplified 20.3%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 76.2

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

   5. Simplified 76.2%

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

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 76.2

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

   8. Simplified 76.2%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 68.7%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right) \cdot \color{blue}{im} \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. swap-sqr N/A

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

       8. metadata-eval N/A

          \[\leadsto \frac{\left(\left(re \cdot re\right) \cdot \color{blue}{\frac{1}{36}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(re \cdot re\right)}{re \cdot \frac{1}{6} - \frac{1}{2}} \cdot im \]

       9. metadata-eval N/A

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

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

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

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

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

       12. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot re\right)}{re \cdot \frac{1}{6} - \frac{1}{2}} \cdot im \]

       13. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \left(re \cdot re\right)}{re \cdot \frac{1}{6} - \frac{1}{2}} \cdot im \]

       14. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \color{blue}{\frac{-1}{4}}\right) \cdot \left(re \cdot re\right)}{re \cdot \frac{1}{6} - \frac{1}{2}} \cdot im \]

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

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \frac{-1}{4}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{re \cdot \frac{1}{6} - \frac{1}{2}} \cdot im \]

       16. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \frac{-1}{4}\right) \cdot \left(re \cdot re\right)}{\color{blue}{re \cdot \frac{1}{6} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot im \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \frac{-1}{4}\right) \cdot \left(re \cdot re\right)}{re \cdot \frac{1}{6} + \color{blue}{\frac{-1}{2}}} \cdot im \]

       18. accelerator-lowering-fma.f64 72.1

           \[\leadsto \frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \left(re \cdot re\right)}{\color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}} \cdot im \]

   13. Applied egg-rr 72.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \left(re \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}} \cdot im \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot re, 1\right), \frac{1}{\mathsf{fma}\left(re, re, 1\right) - re}, re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 -0.02)
     (*
      (sin im)
      (fma
       (* (* re re) (fma (* re re) 0.027777777777777776 -0.25))
       (/ 1.0 (fma re 0.16666666666666666 -0.5))
       (+ re 1.0)))
     (if (<= t_0 4e-96)
       t_1
       (if (<= t_0 1.0)
         (*
          (sin im)
          (fma
           (fma re (* re re) 1.0)
           (/ 1.0 (- (fma re re 1.0) re))
           (* re (* re (fma re 0.16666666666666666 0.5)))))
         t_1)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -0.02) {
		tmp = sin(im) * fma(((re * re) * fma((re * re), 0.027777777777777776, -0.25)), (1.0 / fma(re, 0.16666666666666666, -0.5)), (re + 1.0));
	} else if (t_0 <= 4e-96) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im) * fma(fma(re, (re * re), 1.0), (1.0 / (fma(re, re, 1.0) - re)), (re * (re * fma(re, 0.16666666666666666, 0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(sin(im) * fma(Float64(Float64(re * re) * fma(Float64(re * re), 0.027777777777777776, -0.25)), Float64(1.0 / fma(re, 0.16666666666666666, -0.5)), Float64(re + 1.0)));
	elseif (t_0 <= 4e-96)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) * fma(fma(re, Float64(re * re), 1.0), Float64(1.0 / Float64(fma(re, re, 1.0) - re)), Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-96], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / N[(N[(re * re + 1.0), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 83.2

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

   5. Simplified 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. flip-+ N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

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

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

   7. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)} \cdot \sin im \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   if 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. flip3-+ N/A

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

      7. div-inv N/A

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

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

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

      9. cube-mult N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. associate-+r- N/A

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

      17. --lowering--.f64 N/A

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

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

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

      19. *-commutative N/A

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot re, 1\right), \frac{1}{\mathsf{fma}\left(re, re, 1\right) - re}, re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)} \cdot \sin im \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot re, 1\right), \frac{1}{\mathsf{fma}\left(re, re, 1\right) - re}, re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), t\_0, re + 1\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, t\_0, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma re 0.16666666666666666 -0.5)))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 -0.02)
     (*
      (sin im)
      (fma
       (* (* re re) (fma (* re re) 0.027777777777777776 -0.25))
       t_0
       (+ re 1.0)))
     (if (<= t_1 4e-96)
       t_2
       (if (<= t_1 1.0)
         (* (sin im) (fma (* (* re re) -0.25) t_0 (+ re 1.0)))
         t_2)))))

C

double code(double re, double im) {
	double t_0 = 1.0 / fma(re, 0.16666666666666666, -0.5);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sin(im) * fma(((re * re) * fma((re * re), 0.027777777777777776, -0.25)), t_0, (re + 1.0));
	} else if (t_1 <= 4e-96) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = sin(im) * fma(((re * re) * -0.25), t_0, (re + 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(1.0 / fma(re, 0.16666666666666666, -0.5))
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(sin(im) * fma(Float64(Float64(re * re) * fma(Float64(re * re), 0.027777777777777776, -0.25)), t_0, Float64(re + 1.0)));
	elseif (t_1 <= 4e-96)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(im) * fma(Float64(Float64(re * re) * -0.25), t_0, Float64(re + 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 / N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-96], t$95$2, If[LessEqual[t$95$1, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * t$95$0 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 83.2

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

   5. Simplified 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. flip-+ N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right), \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}, re + 1\right)} \cdot \sin im \]

   7. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)} \cdot \sin im \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999996e-96 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 95.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   if 3.9999999999999996e-96 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 99.8

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot re + 1 \cdot re\right)} + 1\right) \cdot \sin im \]

      2. *-lft-identity N/A

         \[\leadsto \left(\left(\left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot re + \color{blue}{re}\right) + 1\right) \cdot \sin im \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot re + \left(re + 1\right)\right)} \cdot \sin im \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right)} + \left(re + 1\right)\right) \cdot \sin im \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)} + \left(re + 1\right)\right) \cdot \sin im \]

      6. flip-+ N/A

         \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\frac{\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}}{re \cdot \frac{1}{6} - \frac{1}{2}}} + \left(re + 1\right)\right) \cdot \sin im \]

      7. div-inv N/A

         \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}\right)} + \left(re + 1\right)\right) \cdot \sin im \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}} + \left(re + 1\right)\right) \cdot \sin im \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right), \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}, re + 1\right)} \cdot \sin im \]

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)} \cdot \sin im \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot {re}^{2}}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4}, \frac{1}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re + 1\right) \cdot \sin im \]

      4. *-lowering-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right) \cdot \sin im \]

   10. Simplified 99.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right) \cdot -0.25}, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right) \cdot \sin im \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-96}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, re\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 0.0)
     (* im (* im (* im (fma re -0.16666666666666666 -0.16666666666666666))))
     (if (<= t_0 0.0002)
       (fma im (fma (* im im) -0.16666666666666666 re) im)
       (* im (* re (* re (fma re 0.16666666666666666 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = im * (im * (im * fma(re, -0.16666666666666666, -0.16666666666666666)));
	} else if (t_0 <= 0.0002) {
		tmp = fma(im, fma((im * im), -0.16666666666666666, re), im);
	} else {
		tmp = im * (re * (re * fma(re, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666))));
	elseif (t_0 <= 0.0002)
		tmp = fma(im, fma(Float64(im * im), -0.16666666666666666, re), im);
	else
		tmp = Float64(im * Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0002], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

      2. +-lowering-+.f64 44.4

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 44.4%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + im \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(1 + re\right)}\right) + im \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + re\right)\right) + im \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right) + im \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right) + \color{blue}{im} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right), im\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) + re}, im\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{6} \cdot \left(1 + re\right), re\right)}, im\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot \color{blue}{\left(re + 1\right)}, re\right), im\right) \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{re \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, re\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, re\right), im\right) \]

      15. accelerator-lowering-fma.f64 25.2

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}, re\right), im\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)}\right) \]

       9. sub-neg N/A

          \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot re + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(\color{blue}{re \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       12. accelerator-lowering-fma.f64 14.9

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}\right)\right) \]

   11. Simplified 14.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

      2. +-lowering-+.f64 98.3

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + im \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(1 + re\right)}\right) + im \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + re\right)\right) + im \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right) + im \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right) + \color{blue}{im} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right), im\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) + re}, im\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{6} \cdot \left(1 + re\right), re\right)}, im\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot \color{blue}{\left(re + 1\right)}, re\right), im\right) \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{re \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, re\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, re\right), im\right) \]

      15. accelerator-lowering-fma.f64 98.4

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}, re\right), im\right) \]

   8. Simplified 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{6}}, re\right), im\right) \]
   10. Step-by-step derivation

   11. Simplified 98.4%

       \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{-0.16666666666666666}, re\right), im\right) \]

   if 2.0000000000000001e-4 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 89.2

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)} \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right)\right) \cdot \sin im \]

      5. distribute-rgt-in N/A

         \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}\right) \cdot \sin im \]

      6. associate-*l* N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      7. lft-mult-inverse N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      8. metadata-eval N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      12. +-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right) \cdot \sin im \]

      13. *-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot \sin im \]

      14. accelerator-lowering-fma.f64 36.8

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \cdot \sin im \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \]

       6. +-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right)\right) \]

       8. accelerator-lowering-fma.f64 33.0

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right)\right) \]

   11. Simplified 33.0%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 30.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 0.0)
     (* im (* im (* im (fma re -0.16666666666666666 -0.16666666666666666))))
     (if (<= t_0 0.98)
       (* im (fma re (fma re 0.5 1.0) 1.0))
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = im * (im * (im * fma(re, -0.16666666666666666, -0.16666666666666666)));
	} else if (t_0 <= 0.98) {
		tmp = im * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666))));
	elseif (t_0 <= 0.98)
		tmp = Float64(im * fma(re, fma(re, 0.5, 1.0), 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.98], N[(im * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

      2. +-lowering-+.f64 44.4

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 44.4%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + im \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(1 + re\right)}\right) + im \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + re\right)\right) + im \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right) + im \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right) + \color{blue}{im} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right), im\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) + re}, im\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{6} \cdot \left(1 + re\right), re\right)}, im\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot \color{blue}{\left(re + 1\right)}, re\right), im\right) \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{re \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, re\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, re\right), im\right) \]

      15. accelerator-lowering-fma.f64 25.2

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}, re\right), im\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)}\right) \]

       9. sub-neg N/A

          \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot re + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(\color{blue}{re \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       12. accelerator-lowering-fma.f64 14.9

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}\right)\right) \]

   11. Simplified 14.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.97999999999999998

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + \frac{1}{2} \cdot re, 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot re + 1}, 1\right) \cdot \sin im \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{2}} + 1, 1\right) \cdot \sin im \]

      5. accelerator-lowering-fma.f64 98.9

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \cdot \sin im \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 54.7%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{im} \]

   if 0.97999999999999998 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 80.4

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)} \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right)\right) \cdot \sin im \]

      5. distribute-rgt-in N/A

         \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}\right) \cdot \sin im \]

      6. associate-*l* N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      7. lft-mult-inverse N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      8. metadata-eval N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      12. +-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right) \cdot \sin im \]

      13. *-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot \sin im \]

      14. accelerator-lowering-fma.f64 63.4

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 63.4%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \cdot \sin im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \color{blue}{im} \]
   10. Step-by-step derivation

   11. Simplified 57.4%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right) \cdot \color{blue}{im} \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot im \]
   13. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)}\right) \cdot im \]

       2. unpow2 N/A

          \[\leadsto \left(\frac{1}{6} \cdot \left(re \cdot \color{blue}{{re}^{2}}\right)\right) \cdot im \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot re\right) \cdot {re}^{2}\right)} \cdot im \]

       4. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \frac{1}{6}\right)} \cdot {re}^{2}\right) \cdot im \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot im \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot im \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot im \]

       8. associate-*r* N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot re\right) \cdot re\right)}\right) \cdot im \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)}\right) \cdot im \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)}\right) \cdot im \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{6}\right)}\right)\right) \cdot im \]

       12. *-lowering-*.f64 57.3

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \cdot im \]

   14. Simplified 57.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0.98:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* im (* im (* im (fma re -0.16666666666666666 -0.16666666666666666))))
   (*
    (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
    (*
     im
     (fma
      im
      (* im (fma (* im im) 0.008333333333333333 -0.16666666666666666))
      1.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * (im * (im * fma(re, -0.16666666666666666, -0.16666666666666666)));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * (im * fma(im, (im * fma((im * im), 0.008333333333333333, -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(im * Float64(im * Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666))));
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * Float64(im * fma(im, Float64(im * fma(Float64(im * im), 0.008333333333333333, -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

      2. +-lowering-+.f64 44.4

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 44.4%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + im \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(1 + re\right)}\right) + im \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + re\right)\right) + im \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right) + im \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right) + \color{blue}{im} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right), im\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) + re}, im\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{6} \cdot \left(1 + re\right), re\right)}, im\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot \color{blue}{\left(re + 1\right)}, re\right), im\right) \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{re \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, re\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, re\right), im\right) \]

      15. accelerator-lowering-fma.f64 25.2

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}, re\right), im\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)}\right) \]

       9. sub-neg N/A

          \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot re + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(\color{blue}{re \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       12. accelerator-lowering-fma.f64 14.9

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}\right)\right) \]

   11. Simplified 14.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 92.7

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} + 1\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right), 1\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}, 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, 1\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right), 1\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      12. *-lowering-*.f64 57.8

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \]

   8. Simplified 57.8%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 30.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* im (* im (* im (fma re -0.16666666666666666 -0.16666666666666666))))
   (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * (im * (im * fma(re, -0.16666666666666666, -0.16666666666666666)));
	} else {
		tmp = im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(im * Float64(im * Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666))));
	else
		tmp = Float64(im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

      2. +-lowering-+.f64 44.4

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 44.4%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + im \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(1 + re\right)}\right) + im \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + re\right)\right) + im \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right) + im \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right) + \color{blue}{im} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right), im\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) + re}, im\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{6} \cdot \left(1 + re\right), re\right)}, im\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot \color{blue}{\left(re + 1\right)}, re\right), im\right) \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{re \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, re\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, re\right), im\right) \]

      15. accelerator-lowering-fma.f64 25.2

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}, re\right), im\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot re - \frac{1}{6}\right)\right)}\right) \]

       9. sub-neg N/A

          \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot re + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(\color{blue}{re \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       12. accelerator-lowering-fma.f64 14.9

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}\right)\right) \]

   11. Simplified 14.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 92.7

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 55.7%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0002:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0002)
   (* im (fma im (* im -0.16666666666666666) 1.0))
   (* im (* re (* re (* re 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0002) {
		tmp = im * fma(im, (im * -0.16666666666666666), 1.0);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0002)
		tmp = Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0002], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 53.6

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 53.6%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto im \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, 1\right)} \]

      7. *-lowering-*.f64 38.0

         \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)} \]

   if 2.0000000000000001e-4 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 89.2

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)} \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right)\right) \cdot \sin im \]

      5. distribute-rgt-in N/A

         \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}\right) \cdot \sin im \]

      6. associate-*l* N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      7. lft-mult-inverse N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      8. metadata-eval N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      12. +-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right) \cdot \sin im \]

      13. *-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot \sin im \]

      14. accelerator-lowering-fma.f64 36.8

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \cdot \sin im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \color{blue}{im} \]
   10. Step-by-step derivation

   11. Simplified 33.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right) \cdot \color{blue}{im} \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot im \]
   13. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)}\right) \cdot im \]

       2. unpow2 N/A

          \[\leadsto \left(\frac{1}{6} \cdot \left(re \cdot \color{blue}{{re}^{2}}\right)\right) \cdot im \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot re\right) \cdot {re}^{2}\right)} \cdot im \]

       4. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \frac{1}{6}\right)} \cdot {re}^{2}\right) \cdot im \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot im \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot im \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot im \]

       8. associate-*r* N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot re\right) \cdot re\right)}\right) \cdot im \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)}\right) \cdot im \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)}\right) \cdot im \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{6}\right)}\right)\right) \cdot im \]

       12. *-lowering-*.f64 32.9

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \cdot im \]

   14. Simplified 32.9%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0002:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* im (fma im (* im -0.16666666666666666) 1.0))
   (* im (fma re (fma re 0.5 1.0) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * fma(im, (im * -0.16666666666666666), 1.0);
	} else {
		tmp = im * fma(re, fma(re, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im * fma(re, fma(re, 0.5, 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 44.4

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 44.4%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto im \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, 1\right)} \]

      7. *-lowering-*.f64 25.6

         \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + \frac{1}{2} \cdot re, 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot re + 1}, 1\right) \cdot \sin im \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{2}} + 1, 1\right) \cdot \sin im \]

      5. accelerator-lowering-fma.f64 88.4

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \cdot \sin im \]

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 53.5%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0002:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0002)
   (* im (fma im (* im -0.16666666666666666) 1.0))
   (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0002) {
		tmp = im * fma(im, (im * -0.16666666666666666), 1.0);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0002)
		tmp = Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0002], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 53.6

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 53.6%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto im \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, 1\right)} \]

      7. *-lowering-*.f64 38.0

         \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)} \]

   if 2.0000000000000001e-4 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 89.2

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)} \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right)\right) \cdot \sin im \]

      5. distribute-rgt-in N/A

         \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}\right) \cdot \sin im \]

      6. associate-*l* N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      7. lft-mult-inverse N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      8. metadata-eval N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      12. +-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right) \cdot \sin im \]

      13. *-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot \sin im \]

      14. accelerator-lowering-fma.f64 36.8

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \cdot \sin im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \color{blue}{im} \]
   10. Step-by-step derivation

   11. Simplified 33.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right) \cdot \color{blue}{im} \]

   12. Taylor expanded in re around 0

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot im \]
   13. Step-by-step derivation

   14. Simplified 29.9%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{0.5}\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0002:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.98:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.98) im (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.98) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 0.98d0) then
        tmp = im
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.98) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.98:
		tmp = im
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.98)
		tmp = im;
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.98)
		tmp = im;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.98], im, N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.97999999999999998

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.2

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 31.9%

      \[\leadsto \color{blue}{im} \]

   if 0.97999999999999998 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 80.4

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)} \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right)\right) \cdot \sin im \]

      5. distribute-rgt-in N/A

         \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}\right) \cdot \sin im \]

      6. associate-*l* N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      7. lft-mult-inverse N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      8. metadata-eval N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      12. +-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right) \cdot \sin im \]

      13. *-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot \sin im \]

      14. accelerator-lowering-fma.f64 63.4

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 63.4%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \cdot \sin im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \color{blue}{im} \]
   10. Step-by-step derivation

   11. Simplified 57.4%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right) \cdot \color{blue}{im} \]

   12. Taylor expanded in re around 0

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot im \]
   13. Step-by-step derivation

   14. Simplified 51.6%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{0.5}\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.98:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.98:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.98) im (* 0.5 (* re (* re im)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.98) {
		tmp = im;
	} else {
		tmp = 0.5 * (re * (re * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 0.98d0) then
        tmp = im
    else
        tmp = 0.5d0 * (re * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.98) {
		tmp = im;
	} else {
		tmp = 0.5 * (re * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.98:
		tmp = im
	else:
		tmp = 0.5 * (re * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.98)
		tmp = im;
	else
		tmp = Float64(0.5 * Float64(re * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.98)
		tmp = im;
	else
		tmp = 0.5 * (re * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.98], im, N[(0.5 * N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.97999999999999998

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.2

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 31.9%

      \[\leadsto \color{blue}{im} \]

   if 0.97999999999999998 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 80.4

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot \sin im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)} \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right)\right) \cdot \sin im \]

      5. distribute-rgt-in N/A

         \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}\right) \cdot \sin im \]

      6. associate-*l* N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      7. lft-mult-inverse N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      8. metadata-eval N/A

         \[\leadsto \left({re}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot \sin im \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot \sin im \]

      12. +-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}\right) \cdot \sin im \]

      13. *-commutative N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot \sin im \]

      14. accelerator-lowering-fma.f64 63.4

          \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 63.4%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \cdot \sin im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \color{blue}{im} \]
   10. Step-by-step derivation

   11. Simplified 57.4%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right) \cdot \color{blue}{im} \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) \]

       4. associate-*l* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(re \cdot im\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left(im \cdot re\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(im \cdot re\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{\left(re \cdot im\right)}\right) \]

       8. *-lowering-*.f64 40.8

          \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(re \cdot im\right)}\right) \]

   14. Simplified 40.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(re \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 95.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right) + \left(-1 - re\right)\\ \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sin im \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot t\_0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* (fma re 0.16666666666666666 0.5) (* re re)) (- -1.0 re))))
   (if (<= re -0.0042)
     (* (exp re) im)
     (if (<= re 2.05e+102)
       (/
        (*
         (sin im)
         (* (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0) t_0))
        t_0)
       (* (sin im) (* 0.16666666666666666 (* re (* re re))))))))

C

double code(double re, double im) {
	double t_0 = (fma(re, 0.16666666666666666, 0.5) * (re * re)) + (-1.0 - re);
	double tmp;
	if (re <= -0.0042) {
		tmp = exp(re) * im;
	} else if (re <= 2.05e+102) {
		tmp = (sin(im) * (fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * t_0)) / t_0;
	} else {
		tmp = sin(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)) + Float64(-1.0 - re))
	tmp = 0.0
	if (re <= -0.0042)
		tmp = Float64(exp(re) * im);
	elseif (re <= 2.05e+102)
		tmp = Float64(Float64(sin(im) * Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * t_0)) / t_0);
	else
		tmp = Float64(sin(im) * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0042], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 2.05e+102], N[(N[(N[Sin[im], $MachinePrecision] * N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00419999999999999974

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   if -0.00419999999999999974 < re < 2.05e102

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 87.9

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot re + 1 \cdot re\right)} + 1\right) \cdot \sin im \]

      2. *-lft-identity N/A

         \[\leadsto \left(\left(\left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot re + \color{blue}{re}\right) + 1\right) \cdot \sin im \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot re + \left(re + 1\right)\right)} \cdot \sin im \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)\right)} + \left(re + 1\right)\right) \cdot \sin im \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \frac{1}{6} + \frac{1}{2}\right)} + \left(re + 1\right)\right) \cdot \sin im \]

      6. flip-+ N/A

         \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\frac{\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}}{re \cdot \frac{1}{6} - \frac{1}{2}}} + \left(re + 1\right)\right) \cdot \sin im \]

      7. div-inv N/A

         \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{\left(\left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}\right)} + \left(re + 1\right)\right) \cdot \sin im \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(re \cdot re\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}} + \left(re + 1\right)\right) \cdot \sin im \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right), \frac{1}{re \cdot \frac{1}{6} - \frac{1}{2}}, re + 1\right)} \cdot \sin im \]

   7. Applied egg-rr 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)} \cdot \sin im \]

   9. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) - \left(re + 1\right)\right)\right) \cdot \sin im}{\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) - \left(re + 1\right)}} \]

   if 2.05e102 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \sin im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \sin im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \sin im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \sin im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \sin im \]

      7. accelerator-lowering-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot \sin im \]

      2. cube-mult N/A

         \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)}\right) \cdot \sin im \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} \cdot \left(re \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \sin im \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot {re}^{2}\right)}\right) \cdot \sin im \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \sin im \]

      6. *-lowering-*.f64 100.0

         \[\leadsto \left(0.16666666666666666 \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \sin im \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)} \cdot \sin im \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sin im \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right) + \left(-1 - re\right)\right)\right)}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right) + \left(-1 - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 27.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.85 \cdot 10^{+23}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 2.85e+23) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.85e+23) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.85d+23) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.85e+23) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.85e+23:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.85e+23)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.85e+23)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.85e+23], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.85e23

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 54.0

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 35.3%

      \[\leadsto \color{blue}{im} \]

   if 2.85e23 < im

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

      2. +-lowering-+.f64 55.1

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 55.1%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) + im \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(1 + re\right)}\right) + im \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(re + \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + re\right)\right) + im \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(re + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right) + im \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right) + \color{blue}{im} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right), im\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) + re}, im\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{6} \cdot \left(1 + re\right), re\right)}, im\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot \left(1 + re\right), re\right), im\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot \color{blue}{\left(re + 1\right)}, re\right), im\right) \]

      13. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{re \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, re\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, re \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, re\right), im\right) \]

      15. accelerator-lowering-fma.f64 8.7

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right)}, re\right), im\right) \]

   8. Simplified 8.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + re\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(re + 1\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{re \cdot im + 1 \cdot im} \]

       3. *-lft-identity N/A

          \[\leadsto re \cdot im + \color{blue}{im} \]

       4. accelerator-lowering-fma.f64 12.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, im, im\right)} \]

   11. Simplified 12.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{im \cdot re} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot im} \]

       2. *-lowering-*.f64 14.1

          \[\leadsto \color{blue}{re \cdot im} \]

   14. Simplified 14.1%

       \[\leadsto \color{blue}{re \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 29.3% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(im, re, im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma im re im))

C

double code(double re, double im) {
	return fma(im, re, im);
}

Julia

function code(re, im)
	return fma(im, re, im)
end

Wolfram

code[re_, im_] := N[(im * re + im), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto e^{re} \cdot \color{blue}{im} \]
4. Step-by-step derivation

5. Simplified 68.9%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im + im \cdot re} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{im \cdot re + im} \]

   2. accelerator-lowering-fma.f64 32.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

8. Simplified 32.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]
9. Add Preprocessing

Alternative 23: 26.1% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

double code(double re, double im) {
	return im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im
end function

Java

public static double code(double re, double im) {
	return im;
}

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

function tmp = code(re, im)
	tmp = im;
end

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\sin im} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 54.0

      \[\leadsto \color{blue}{\sin im} \]

5. Simplified 54.0%

   \[\leadsto \color{blue}{\sin im} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 28.0%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))