math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 21.5s

Alternatives: 26

Speedup: 1.0×

• 23.9% 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: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \sin im \cdot \left(re + 1\right)\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
     (*
      (* im (* re (* re re)))
      (fma (* im im) -0.027777777777777776 0.16666666666666666))
     (if (<= t_0 -0.02)
       t_1
       (if (<= t_0 1e-55) t_2 (if (<= t_0 50000.0) 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 = (im * (re * (re * re))) * fma((im * im), -0.027777777777777776, 0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 1e-55) {
		tmp = t_2;
	} else if (t_0 <= 50000.0) {
		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(im * Float64(re * Float64(re * re))) * fma(Float64(im * im), -0.027777777777777776, 0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 1e-55)
		tmp = t_2;
	elseif (t_0 <= 50000.0)
		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[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 1e-55], t$95$2, If[LessEqual[t$95$0, 50000.0], 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 78.1

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

   5. Simplified 78.1%

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

   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 78.1

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

   8. Simplified 78.1%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{36} \cdot \left({im}^{2} \cdot {re}^{3}\right) + \frac{1}{6} \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{36} \cdot \left({im}^{2} \cdot {re}^{3}\right)\right) + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{-1}{36} \cdot \color{blue}{\left({re}^{3} \cdot {im}^{2}\right)}\right) + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       3. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{36} \cdot {re}^{3}\right) \cdot {im}^{2}\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{36} \cdot {re}^{3}\right) \cdot {im}^{2}\right) \cdot im} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot \left({re}^{3} \cdot {im}^{2}\right)\right)} \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\frac{-1}{36} \cdot \color{blue}{\left({im}^{2} \cdot {re}^{3}\right)}\right) \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       7. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       8. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left({re}^{3} \cdot im\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot {re}^{3}\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       10. associate-*r* N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\left(im \cdot \frac{1}{6}\right) \cdot {re}^{3}} \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\left(\frac{1}{6} \cdot im\right)} \cdot {re}^{3} \]

       12. associate-*r* N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]

       13. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left(im \cdot {re}^{3}\right) \cdot \left(\frac{-1}{36} \cdot {im}^{2} + \frac{1}{6}\right)} \]

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

           \[\leadsto \color{blue}{\left(im \cdot {re}^{3}\right) \cdot \left(\frac{-1}{36} \cdot {im}^{2} + \frac{1}{6}\right)} \]

   11. Simplified 58.1%

       \[\leadsto \color{blue}{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 9.99999999999999995e-56 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   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 97.2

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

   5. Simplified 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999995e-56 or 5e4 < (*.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 93.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-55}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* im (* re (* re re)))
      (fma (* im im) -0.027777777777777776 0.16666666666666666))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 1e-55) t_1 (if (<= t_0 50000.0) (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 = (im * (re * (re * re))) * fma((im * im), -0.027777777777777776, 0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 1e-55) {
		tmp = t_1;
	} else if (t_0 <= 50000.0) {
		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(im * Float64(re * Float64(re * re))) * fma(Float64(im * im), -0.027777777777777776, 0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 1e-55)
		tmp = t_1;
	elseif (t_0 <= 50000.0)
		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[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-55], t$95$1, If[LessEqual[t$95$0, 50000.0], 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 78.1

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

   5. Simplified 78.1%

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

   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 78.1

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

   8. Simplified 78.1%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{36} \cdot \left({im}^{2} \cdot {re}^{3}\right) + \frac{1}{6} \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{36} \cdot \left({im}^{2} \cdot {re}^{3}\right)\right) + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{-1}{36} \cdot \color{blue}{\left({re}^{3} \cdot {im}^{2}\right)}\right) + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       3. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{36} \cdot {re}^{3}\right) \cdot {im}^{2}\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{36} \cdot {re}^{3}\right) \cdot {im}^{2}\right) \cdot im} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot \left({re}^{3} \cdot {im}^{2}\right)\right)} \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\frac{-1}{36} \cdot \color{blue}{\left({im}^{2} \cdot {re}^{3}\right)}\right) \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       7. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       8. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left({re}^{3} \cdot im\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot {re}^{3}\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       10. associate-*r* N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\left(im \cdot \frac{1}{6}\right) \cdot {re}^{3}} \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\left(\frac{1}{6} \cdot im\right)} \cdot {re}^{3} \]

       12. associate-*r* N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]

       13. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left(im \cdot {re}^{3}\right) \cdot \left(\frac{-1}{36} \cdot {im}^{2} + \frac{1}{6}\right)} \]

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

           \[\leadsto \color{blue}{\left(im \cdot {re}^{3}\right) \cdot \left(\frac{-1}{36} \cdot {im}^{2} + \frac{1}{6}\right)} \]

   11. Simplified 58.1%

       \[\leadsto \color{blue}{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 9.99999999999999995e-56 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 96.0

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

   5. Simplified 96.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999995e-56 or 5e4 < (*.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 93.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-55}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-55}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, t\_1, -1\right)}{im}}\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{re + -1}{im \cdot \mathsf{fma}\left(re, t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma re (fma re 0.16666666666666666 0.5) 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (* im (* re (* re re)))
      (fma (* im im) -0.027777777777777776 0.16666666666666666))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 1e-55)
         (/ -1.0 (/ (fma re t_1 -1.0) im))
         (if (<= t_0 50000.0)
           (sin im)
           (/
            1.0
            (/
             (+ re -1.0)
             (*
              im
              (fma
               re
               (* t_1 (fma (fma re 0.16666666666666666 0.5) (* re re) re))
               -1.0))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (im * (re * (re * re))) * fma((im * im), -0.027777777777777776, 0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 1e-55) {
		tmp = -1.0 / (fma(re, t_1, -1.0) / im);
	} else if (t_0 <= 50000.0) {
		tmp = sin(im);
	} else {
		tmp = 1.0 / ((re + -1.0) / (im * fma(re, (t_1 * fma(fma(re, 0.16666666666666666, 0.5), (re * re), re)), -1.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(im * Float64(re * Float64(re * re))) * fma(Float64(im * im), -0.027777777777777776, 0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 1e-55)
		tmp = Float64(-1.0 / Float64(fma(re, t_1, -1.0) / im));
	elseif (t_0 <= 50000.0)
		tmp = sin(im);
	else
		tmp = Float64(1.0 / Float64(Float64(re + -1.0) / Float64(im * fma(re, Float64(t_1 * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re)), -1.0))));
	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[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-55], N[(-1.0 / N[(N[(re * t$95$1 + -1.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50000.0], N[Sin[im], $MachinePrecision], N[(1.0 / N[(N[(re + -1.0), $MachinePrecision] / N[(im * N[(re * N[(t$95$1 * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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 78.1

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

   5. Simplified 78.1%

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

   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 78.1

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

   8. Simplified 78.1%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{36} \cdot \left({im}^{2} \cdot {re}^{3}\right) + \frac{1}{6} \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{36} \cdot \left({im}^{2} \cdot {re}^{3}\right)\right) + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{-1}{36} \cdot \color{blue}{\left({re}^{3} \cdot {im}^{2}\right)}\right) + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       3. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{36} \cdot {re}^{3}\right) \cdot {im}^{2}\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{36} \cdot {re}^{3}\right) \cdot {im}^{2}\right) \cdot im} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot \left({re}^{3} \cdot {im}^{2}\right)\right)} \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\frac{-1}{36} \cdot \color{blue}{\left({im}^{2} \cdot {re}^{3}\right)}\right) \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       7. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \cdot im + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       8. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left({re}^{3} \cdot im\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot {re}^{3}\right)} + im \cdot \left(\frac{1}{6} \cdot {re}^{3}\right) \]

       10. associate-*r* N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\left(im \cdot \frac{1}{6}\right) \cdot {re}^{3}} \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\left(\frac{1}{6} \cdot im\right)} \cdot {re}^{3} \]

       12. associate-*r* N/A

           \[\leadsto \left(\frac{-1}{36} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{3}\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]

       13. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left(im \cdot {re}^{3}\right) \cdot \left(\frac{-1}{36} \cdot {im}^{2} + \frac{1}{6}\right)} \]

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

           \[\leadsto \color{blue}{\left(im \cdot {re}^{3}\right) \cdot \left(\frac{-1}{36} \cdot {im}^{2} + \frac{1}{6}\right)} \]

   11. Simplified 58.1%

       \[\leadsto \color{blue}{\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 9.99999999999999995e-56 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 96.0

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

   5. Simplified 96.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999995e-56

   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 48.7

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

   5. Simplified 48.7%

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

   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 48.7%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 48.0%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 84.6

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

   13. Simplified 84.6%

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

   if 5e4 < (*.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 65.6

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

   5. Simplified 65.6%

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

   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 41.0%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 12.6%

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

   11. Taylor expanded in re around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-lowering-+.f64 48.8

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

   13. Simplified 48.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-55}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}{im}}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{re + -1}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\frac{\sin im}{\frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 -0.02)
     (/
      (sin im)
      (/ 1.0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
     (if (<= t_0 2e-130)
       t_1
       (if (<= t_0 50000.0)
         (*
          (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 <= -0.02) {
		tmp = sin(im) / (1.0 / fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (t_0 <= 2e-130) {
		tmp = t_1;
	} else if (t_0 <= 50000.0) {
		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 <= -0.02)
		tmp = Float64(sin(im) / Float64(1.0 / fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (t_0 <= 2e-130)
		tmp = t_1;
	elseif (t_0 <= 50000.0)
		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, -0.02], N[(N[Sin[im], $MachinePrecision] / N[(1.0 / N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-130], t$95$1, If[LessEqual[t$95$0, 50000.0], 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 3 regimes

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

   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 86.6

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

   5. Simplified 86.6%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

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

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

   7. Applied egg-rr 86.6%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-130 or 5e4 < (*.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 93.1%

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

   if 2.0000000000000002e-130 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   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. 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 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\frac{\sin im}{\frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)}}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-130}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 -0.02)
     (* (sin im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= t_0 2e-130)
       t_1
       (if (<= t_0 50000.0)
         (*
          (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 <= -0.02) {
		tmp = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 2e-130) {
		tmp = t_1;
	} else if (t_0 <= 50000.0) {
		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 <= -0.02)
		tmp = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 2e-130)
		tmp = t_1;
	elseif (t_0 <= 50000.0)
		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, -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, 2e-130], t$95$1, If[LessEqual[t$95$0, 50000.0], 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 3 regimes

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

   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 86.6

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

   5. Simplified 86.6%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-130 or 5e4 < (*.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 93.1%

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

   if 2.0000000000000002e-130 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   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. 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 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-130}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 -0.02)
     t_0
     (if (<= t_1 2e-130) t_2 (if (<= t_1 50000.0) t_0 t_2)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 2e-130) {
		tmp = t_2;
	} else if (t_1 <= 50000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 2e-130)
		tmp = t_2;
	elseif (t_1 <= 50000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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$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], t$95$0, If[LessEqual[t$95$1, 2e-130], t$95$2, If[LessEqual[t$95$1, 50000.0], t$95$0, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 2.0000000000000002e-130 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   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 92.2

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

   5. Simplified 92.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-130 or 5e4 < (*.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 93.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-130}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)) (t_1 (* (exp re) (sin im))))
   (if (<= t_1 -0.02)
     (* (sin im) (fma re (fma re 0.5 1.0) 1.0))
     (if (<= t_1 1e-55)
       t_0
       (if (<= t_1 50000.0)
         (* (sin im) (fma (* re re) 0.5 (+ re 1.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (t_1 <= 1e-55) {
		tmp = t_0;
	} else if (t_1 <= 50000.0) {
		tmp = sin(im) * fma((re * re), 0.5, (re + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_1 <= 1e-55)
		tmp = t_0;
	elseif (t_1 <= 50000.0)
		tmp = Float64(sin(im) * fma(Float64(re * re), 0.5, Float64(re + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-55], t$95$0, If[LessEqual[t$95$1, 50000.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.5 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

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

   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 79.3

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

   5. Simplified 79.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999995e-56 or 5e4 < (*.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 93.3%

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

   if 9.99999999999999995e-56 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   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 100.0

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

   5. Simplified 100.0%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-55}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (sin im) (fma re (fma re 0.5 1.0) 1.0))))
   (if (<= t_1 -0.02)
     t_2
     (if (<= t_1 1e-55) t_0 (if (<= t_1 50000.0) t_2 t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double t_2 = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double tmp;
	if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 1e-55) {
		tmp = t_0;
	} else if (t_1 <= 50000.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 1e-55)
		tmp = t_0;
	elseif (t_1 <= 50000.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 1e-55], t$95$0, If[LessEqual[t$95$1, 50000.0], t$95$2, t$95$0]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 9.99999999999999995e-56 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e4

   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 87.7

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

   5. Simplified 87.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999995e-56 or 5e4 < (*.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 93.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-55}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 50000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-130}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, t\_1, -1\right)}{im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{re + -1}{im \cdot \mathsf{fma}\left(re, t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma re (fma re 0.16666666666666666 0.5) 1.0)))
   (if (<= t_0 -0.02)
     (* im (* re (* re (fma (* im im) -0.08333333333333333 0.5))))
     (if (<= t_0 2e-130)
       (/ -1.0 (/ (fma re t_1 -1.0) im))
       (/
        1.0
        (/
         (+ re -1.0)
         (*
          im
          (fma
           re
           (* t_1 (fma (fma re 0.16666666666666666 0.5) (* re re) re))
           -1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = im * (re * (re * fma((im * im), -0.08333333333333333, 0.5)));
	} else if (t_0 <= 2e-130) {
		tmp = -1.0 / (fma(re, t_1, -1.0) / im);
	} else {
		tmp = 1.0 / ((re + -1.0) / (im * fma(re, (t_1 * fma(fma(re, 0.16666666666666666, 0.5), (re * re), re)), -1.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(im * Float64(re * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.5))));
	elseif (t_0 <= 2e-130)
		tmp = Float64(-1.0 / Float64(fma(re, t_1, -1.0) / im));
	else
		tmp = Float64(1.0 / Float64(Float64(re + -1.0) / Float64(im * fma(re, Float64(t_1 * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re)), -1.0))));
	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[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(im * N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-130], N[(-1.0 / N[(N[(re * t$95$1 + -1.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(re + -1.0), $MachinePrecision] / N[(im * N[(re * N[(t$95$1 * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 25.0%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

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

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

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

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

       14. distribute-rgt-in N/A

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

       15. metadata-eval N/A

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

       16. +-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-*l* N/A

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

       19. metadata-eval N/A

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

       20. metadata-eval N/A

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

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

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

   11. Simplified 25.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-130

   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 47.3

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

   5. Simplified 47.3%

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

   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 47.3%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 46.6%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 84.2

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

   13. Simplified 84.2%

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

   if 2.0000000000000002e-130 < (*.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 88.5

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

   5. Simplified 88.5%

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

   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 24.0%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 14.5%

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

   11. Taylor expanded in re around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-lowering-+.f64 26.5

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

   13. Simplified 26.5%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-130}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}{im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{re + -1}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.02)
     (* im (* re (* re (fma (* im im) -0.08333333333333333 0.5))))
     (if (<= t_0 0.0)
       (/
        -1.0
        (/ (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) -1.0) im))
       (*
        im
        (fma
         (* (* re re) (fma (* re re) 0.027777777777777776 -0.25))
         (/ 1.0 (fma re 0.16666666666666666 -0.5))
         (+ re 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = im * (re * (re * fma((im * im), -0.08333333333333333, 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), -1.0) / im);
	} else {
		tmp = im * fma(((re * re) * fma((re * re), 0.027777777777777776, -0.25)), (1.0 / fma(re, 0.16666666666666666, -0.5)), (re + 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(im * Float64(re * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.5))));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), -1.0) / im));
	else
		tmp = Float64(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)));
	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, -0.02], N[(im * N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[(im * 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]]]]

Derivation

1. Split input into 3 regimes

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

   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 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 25.0%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

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

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

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

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

       14. distribute-rgt-in N/A

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

       15. metadata-eval N/A

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

       16. +-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-*l* N/A

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

       19. metadata-eval N/A

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

       20. metadata-eval N/A

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

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

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

   11. Simplified 25.3%

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

   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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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 37.0

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

   5. Simplified 37.0%

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

   6. Taylor expanded in 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 37.0%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 36.2%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 82.9

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

   13. Simplified 82.9%

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

   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 89.8

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

   5. Simplified 89.8%

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

   6. 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 38.8%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. flip-+ N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

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

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

   10. Applied egg-rr 38.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, t\_1, -1\right)}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, t\_1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma re (fma re 0.16666666666666666 0.5) 1.0)))
   (if (<= t_0 -0.02)
     (* im (* re (* re (fma (* im im) -0.08333333333333333 0.5))))
     (if (<= t_0 0.0)
       (/ -1.0 (/ (fma re t_1 -1.0) im))
       (* im (fma re t_1 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = im * (re * (re * fma((im * im), -0.08333333333333333, 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (fma(re, t_1, -1.0) / im);
	} else {
		tmp = im * fma(re, t_1, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(im * Float64(re * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.5))));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(fma(re, t_1, -1.0) / im));
	else
		tmp = Float64(im * fma(re, t_1, 1.0));
	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[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(im * N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(N[(re * t$95$1 + -1.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 25.0%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

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

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

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

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

       14. distribute-rgt-in N/A

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

       15. metadata-eval N/A

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

       16. +-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-*l* N/A

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

       19. metadata-eval N/A

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

       20. metadata-eval N/A

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

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

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

   11. Simplified 25.3%

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

   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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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 37.0

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

   5. Simplified 37.0%

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

   6. Taylor expanded in 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 37.0%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 36.2%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 82.9

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

   13. Simplified 82.9%

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

   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 89.8

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

   5. Simplified 89.8%

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

   6. 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 38.8%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}{im}}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(im, re \cdot 0.5, im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 0.0)
     (* (* im (* im -0.16666666666666666)) (fma re im im))
     (if (<= t_0 0.9)
       (fma re (fma im (* re 0.5) im) im)
       (* im (* (fma re 0.16666666666666666 0.5) (* re re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * (im * -0.16666666666666666)) * fma(re, im, im);
	} else if (t_0 <= 0.9) {
		tmp = fma(re, fma(im, (re * 0.5), im), im);
	} else {
		tmp = im * (fma(re, 0.16666666666666666, 0.5) * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * Float64(im * -0.16666666666666666)) * fma(re, im, im));
	elseif (t_0 <= 0.9)
		tmp = fma(re, fma(im, Float64(re * 0.5), im), im);
	else
		tmp = Float64(im * Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)));
	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[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9], N[(re * N[(im * N[(re * 0.5), $MachinePrecision] + im), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 31.9%

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

   9. Taylor expanded in re around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

       15. +-commutative N/A

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

       16. distribute-lft1-in N/A

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

       17. accelerator-lowering-fma.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 11.9

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

   14. Simplified 11.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.900000000000000022

   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 46.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 44.2

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

   8. Simplified 44.2%

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

   if 0.900000000000000022 < (*.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 76.1

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

   5. Simplified 76.1%

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

   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 29.3%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

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

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 30.0

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

   11. Simplified 30.0%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0.9:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(im, re \cdot 0.5, im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(-1 - re\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (- -1.0 re) (* 0.16666666666666666 (* im (* im im))))
   (* 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 = (-1.0 - re) * (0.16666666666666666 * (im * (im * im)));
	} 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(Float64(-1.0 - re) * Float64(0.16666666666666666 * Float64(im * Float64(im * im))));
	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[(N[(-1.0 - re), $MachinePrecision] * N[(0.16666666666666666 * N[(im * N[(im * im), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 31.9%

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

   9. Taylor expanded in re around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

       15. +-commutative N/A

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

       16. distribute-lft1-in N/A

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

       17. accelerator-lowering-fma.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in im around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. mul-1-neg N/A

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

       8. unsub-neg N/A

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

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

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

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

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 15.4

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

   14. Simplified 15.4%

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

   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 89.8

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

   5. Simplified 89.8%

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

   6. 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 38.8%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(-1 - re\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(-1 - re\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (- -1.0 re) (* 0.16666666666666666 (* im (* im im))))
   (* im (fma re (* re (* re 0.16666666666666666)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = (-1.0 - re) * (0.16666666666666666 * (im * (im * im)));
	} else {
		tmp = im * fma(re, (re * (re * 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(Float64(-1.0 - re) * Float64(0.16666666666666666 * Float64(im * Float64(im * im))));
	else
		tmp = Float64(im * fma(re, Float64(re * Float64(re * 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[(N[(-1.0 - re), $MachinePrecision] * N[(0.16666666666666666 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 31.9%

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

   9. Taylor expanded in re around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

       15. +-commutative N/A

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

       16. distribute-lft1-in N/A

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

       17. accelerator-lowering-fma.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in im around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. mul-1-neg N/A

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

       8. unsub-neg N/A

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

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

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

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

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 15.4

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

   14. Simplified 15.4%

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

   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 89.8

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

   5. Simplified 89.8%

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

   6. 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 38.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 37.1

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

   11. Simplified 37.1%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(-1 - re\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* im (* im -0.16666666666666666)) (fma re im im))
   (* im (fma re (* re (* re 0.16666666666666666)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = (im * (im * -0.16666666666666666)) * fma(re, im, im);
	} else {
		tmp = im * fma(re, (re * (re * 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 31.9%

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

   9. Taylor expanded in re around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

       15. +-commutative N/A

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

       16. distribute-lft1-in N/A

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

       17. accelerator-lowering-fma.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 11.9

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

   14. Simplified 11.9%

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

   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 89.8

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

   5. Simplified 89.8%

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

   6. 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 38.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 37.1

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

   11. Simplified 37.1%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* im (* im -0.16666666666666666)) (fma re im im))
   (* 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 * (im * -0.16666666666666666)) * fma(re, im, im);
	} 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(Float64(im * Float64(im * -0.16666666666666666)) * fma(re, im, im));
	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[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * im + im), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 31.9%

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

   9. Taylor expanded in re around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

       15. +-commutative N/A

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

       16. distribute-lft1-in N/A

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

       17. accelerator-lowering-fma.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 11.9

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

   14. Simplified 11.9%

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

   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 87.5

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

   5. Simplified 87.5%

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

   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 38.5%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 28.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* im (* -0.16666666666666666 (* im (fma re im im))))
   (* 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 * (-0.16666666666666666 * (im * fma(re, im, im)));
	} 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 * Float64(-0.16666666666666666 * Float64(im * fma(re, im, im))));
	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[(-0.16666666666666666 * N[(im * N[(re * im + im), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

   8. Simplified 31.9%

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

   9. Taylor expanded in re around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

       15. +-commutative N/A

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

       16. distribute-lft1-in N/A

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

       17. accelerator-lowering-fma.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. distribute-lft-in N/A

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

       15. *-rgt-identity N/A

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

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

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

       17. +-commutative N/A

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

       18. *-commutative N/A

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

       19. accelerator-lowering-fma.f64 11.3

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

   14. Simplified 11.3%

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

   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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 87.5

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

   5. Simplified 87.5%

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

   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 38.5%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.001)
   (fma im (* (* im im) -0.16666666666666666) im)
   (* im (* 0.16666666666666666 (* re (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.001) {
		tmp = fma(im, ((im * im) * -0.16666666666666666), im);
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.001)
		tmp = fma(im, Float64(Float64(im * im) * -0.16666666666666666), im);
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.001], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 48.6

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

   5. Simplified 48.6%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 33.0

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

   8. Simplified 33.0%

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

   if 1e-3 < (*.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 87.3

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

   5. Simplified 87.3%

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

   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 16.8%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 17.3

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

   11. Simplified 17.3%

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

4. Final simplification 28.8%

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

Alternative 20: 36.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.001)
   (fma im (* (* im im) -0.16666666666666666) im)
   (* 0.16666666666666666 (* im (* re (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.001) {
		tmp = fma(im, ((im * im) * -0.16666666666666666), im);
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.001)
		tmp = fma(im, Float64(Float64(im * im) * -0.16666666666666666), im);
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.001], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 48.6

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

   5. Simplified 48.6%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 33.0

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

   8. Simplified 33.0%

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

   if 1e-3 < (*.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 87.3

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   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 16.8%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left({re}^{3} \cdot im\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left({re}^{3} \cdot im\right)} \]

       4. cube-mult N/A

          \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot im\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot im\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right)} \cdot im\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot im\right) \]

       8. *-lowering-*.f64 17.3

          \[\leadsto 0.16666666666666666 \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot im\right) \]

   11. Simplified 17.3%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (fma im (* (* im im) -0.16666666666666666) im)
   (* 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 = fma(im, ((im * im) * -0.16666666666666666), im);
	} 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 = fma(im, Float64(Float64(im * im) * -0.16666666666666666), im);
	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[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + im), $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 41.6

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 41.6%

      \[\leadsto \color{blue}{\sin im} \]

   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. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, \frac{-1}{6} \cdot {im}^{2}, im\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\frac{-1}{6} \cdot {im}^{2}}, im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(im, \frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

      7. *-lowering-*.f64 23.4

         \[\leadsto \mathsf{fma}\left(im, -0.16666666666666666 \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

   8. Simplified 23.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)} \]

   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 87.5

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \cdot \sin im \]

   5. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \cdot \sin im \]

   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 38.5%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 31.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (fma im (* (* im im) -0.16666666666666666) im)
   (fma im re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = fma(im, ((im * im) * -0.16666666666666666), im);
	} else {
		tmp = fma(im, re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = fma(im, Float64(Float64(im * im) * -0.16666666666666666), im);
	else
		tmp = fma(im, re, im);
	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[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], N[(im * re + im), $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 41.6

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 41.6%

      \[\leadsto \color{blue}{\sin im} \]

   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. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, \frac{-1}{6} \cdot {im}^{2}, im\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{\frac{-1}{6} \cdot {im}^{2}}, im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(im, \frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

      7. *-lowering-*.f64 23.4

         \[\leadsto \mathsf{fma}\left(im, -0.16666666666666666 \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

   8. Simplified 23.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)} \]

   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 im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 46.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   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 32.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

   8. Simplified 32.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 95.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00076:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00076)
     t_0
     (if (<= re 9.5e-6)
       (* (sin im) (+ re 1.0))
       (if (<= re 1.9e+154) t_0 (* (sin im) (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00076) {
		tmp = t_0;
	} else if (re <= 9.5e-6) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.00076d0)) then
        tmp = t_0
    else if (re <= 9.5d-6) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = t_0
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.00076) {
		tmp = t_0;
	} else if (re <= 9.5e-6) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.00076:
		tmp = t_0
	elif re <= 9.5e-6:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = t_0
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.00076)
		tmp = t_0;
	elseif (re <= 9.5e-6)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.00076)
		tmp = t_0;
	elseif (re <= 9.5e-6)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.00076], t$95$0, If[LessEqual[re, 9.5e-6], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.6000000000000004e-4 or 9.5000000000000005e-6 < re < 1.8999999999999999e154

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
   4. Step-by-step derivation

   5. Simplified 92.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   if -7.6000000000000004e-4 < re < 9.5000000000000005e-6

   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 99.7

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 100.0

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \cdot \sin im \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \cdot \sin im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)} \cdot \sin im \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \sin im \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot re\right) \cdot re\right)} \cdot \sin im \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{2}\right)}\right) \cdot \sin im \]

      6. *-lowering-*.f64 100.0

         \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot 0.5\right)}\right) \cdot \sin im \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \cdot \sin im \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00076:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.6 \cdot 10^{+46}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7.6e+46) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.6e+46) {
		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 <= 7.6d+46) 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 <= 7.6e+46) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.6e+46:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.6e+46)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.6e+46)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.6e+46], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.5999999999999998e46

   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 51.0

         \[\leadsto \color{blue}{\sin im} \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 30.6%

      \[\leadsto \color{blue}{im} \]

   if 7.5999999999999998e46 < 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 69.8

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   6. 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 12.5%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + im \cdot re} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot re + im} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot im} + im \]

       3. accelerator-lowering-fma.f64 10.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, im, im\right)} \]

   11. Simplified 10.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, im, im\right)} \]

   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 11.6

          \[\leadsto \color{blue}{re \cdot im} \]

   14. Simplified 11.6%

       \[\leadsto \color{blue}{re \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 30.6% 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 65.0%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

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 28.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

8. Simplified 28.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]
9. Add Preprocessing

Alternative 26: 27.6% 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 52.5

      \[\leadsto \color{blue}{\sin im} \]

5. Simplified 52.5%

   \[\leadsto \color{blue}{\sin im} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 24.3%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))