math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 18.3s

Alternatives: 20

Speedup: 1.0×

• 24.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot t\_0\right)\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 10^{-78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re 0.5 1.0) 1.0))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 (- INFINITY))
     (* im (* (fma im (* im -0.16666666666666666) 1.0) t_0))
     (if (<= t_1 -0.1)
       (sin im)
       (if (<= t_1 1e-78) t_2 (if (<= t_1 1.0) (* (sin im) t_0) t_2))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, 0.5, 1.0), 1.0);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = im * (fma(im, (im * -0.16666666666666666), 1.0) * t_0);
	} else if (t_1 <= -0.1) {
		tmp = sin(im);
	} else if (t_1 <= 1e-78) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = sin(im) * t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, 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 <= Float64(-Inf))
		tmp = Float64(im * Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * t_0));
	elseif (t_1 <= -0.1)
		tmp = sin(im);
	elseif (t_1 <= 1e-78)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(im) * t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $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, (-Infinity)], N[(im * N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 1e-78], t$95$2, If[LessEqual[t$95$1, 1.0], N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$2]]]]]]]

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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-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. lower-fma.f64 48.1

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

   5. Applied rewrites 48.1%

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

   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. lower-*.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. lower-*.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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative 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. lower-*.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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. lower-sin.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999999e-79 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 9.99999999999999999e-79 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-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. lower-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. Applied rewrites 100.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-78}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      im
      (*
       (fma im (* im -0.16666666666666666) 1.0)
       (fma re (fma re 0.5 1.0) 1.0)))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 1e-78)
         t_1
         (if (<= t_0 1.0) (* (sin im) (+ re 1.0)) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im * (fma(im, (im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 1e-78) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im * Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 1e-78)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(im * N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-78], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-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. lower-fma.f64 48.1

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

   5. Applied rewrites 48.1%

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

   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. lower-*.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. lower-*.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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative 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. lower-*.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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. lower-sin.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999999e-79 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 9.99999999999999999e-79 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-78}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      im
      (*
       (fma im (* im -0.16666666666666666) 1.0)
       (fma re (fma re 0.5 1.0) 1.0)))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 2e-74) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im * (fma(im, (im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 2e-74) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im * Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 2e-74)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(im * N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-74], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-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. lower-fma.f64 48.1

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

   5. Applied rewrites 48.1%

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

   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. lower-*.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. lower-*.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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative 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. lower-*.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 1.99999999999999992e-74 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999992e-74 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-74}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, im\right), im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      im
      (*
       (fma im (* im -0.16666666666666666) 1.0)
       (fma re (fma re 0.5 1.0) 1.0)))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 0.0)
         (* im (* re (* 0.008333333333333333 (* (* im im) (* im im)))))
         (if (<= t_0 1.0)
           (sin im)
           (fma
            re
            (fma
             (* re im)
             (/ (fma (* re (* re re)) 0.004629629629629629 0.125) 0.25)
             im)
            im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im * (fma(im, (im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0));
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = im * (re * (0.008333333333333333 * ((im * im) * (im * im))));
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = fma(re, fma((re * im), (fma((re * (re * re)), 0.004629629629629629, 0.125) / 0.25), im), im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im * Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0)));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(im * Float64(re * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im)))));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = fma(re, fma(Float64(re * im), Float64(fma(Float64(re * Float64(re * re)), 0.004629629629629629, 0.125) / 0.25), im), im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(im * N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im * N[(re * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(re * N[(N[(re * im), $MachinePrecision] * N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + im), $MachinePrecision] + im), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-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. lower-fma.f64 48.1

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

   5. Applied rewrites 48.1%

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

   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. lower-*.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. lower-*.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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative 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. lower-*.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 3.9

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

   11. Applied rewrites 3.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({im}^{4} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {im}^{4}\right) \cdot re\right)} \]

       2. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {im}^{4}\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {im}^{4}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

       11. lower-*.f64 25.4

           \[\leadsto im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

   14. Applied rewrites 25.4%

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 96.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 48.4

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

   8. Applied rewrites 48.4%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-fma.f64 N/A

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

      5. cube-mult N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \color{blue}{\frac{1}{216}}, {\frac{1}{2}}^{3}\right)}{\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, im\right), im\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \color{blue}{\frac{1}{8}}\right)}{\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, im\right), im\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}}, im\right), im\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}}, im\right), im\right) \]

      12. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\left(\color{blue}{\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)} + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\left(\left(re \cdot re\right) \cdot \color{blue}{\frac{1}{36}} + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\left(\left(re \cdot re\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\mathsf{fma}\left(re \cdot re, \frac{-1}{6} \cdot \frac{-1}{6}, \frac{1}{2} \cdot \frac{1}{2}\right)} - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{6} \cdot \frac{-1}{6}, \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{36}}, \frac{1}{2} \cdot \frac{1}{2}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \color{blue}{\frac{1}{4}}\right) - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, im\right), im\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \frac{1}{4}\right) - \color{blue}{re \cdot \left(\frac{1}{6} \cdot \frac{1}{2}\right)}}, im\right), im\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(re \cdot re, \frac{1}{36}, \frac{1}{4}\right) - \color{blue}{re \cdot \left(\frac{1}{6} \cdot \frac{1}{2}\right)}}, im\right), im\right) \]

      21. metadata-eval 4.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot \color{blue}{0.08333333333333333}}, im\right), im\right) \]

   10. Applied rewrites 4.0%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \color{blue}{\frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot 0.08333333333333333}}, im\right), im\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, im\right), im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.2%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(im \cdot re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, im\right), im\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, im\right), im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right), \mathsf{fma}\left(re, im, im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.1)
     (*
      im
      (*
       (fma im (* im -0.16666666666666666) 1.0)
       (fma re (fma re 0.5 1.0) 1.0)))
     (if (<= t_0 0.0)
       (* im (* re (* 0.008333333333333333 (* (* im im) (* im im)))))
       (fma
        im
        (* (fma re 0.16666666666666666 0.5) (* re re))
        (fma re im im))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = im * (fma(im, (im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0));
	} else if (t_0 <= 0.0) {
		tmp = im * (re * (0.008333333333333333 * ((im * im) * (im * im))));
	} else {
		tmp = fma(im, (fma(re, 0.16666666666666666, 0.5) * (re * re)), fma(re, im, im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(im * Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(im * Float64(re * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im)))));
	else
		tmp = fma(im, Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)), fma(re, im, im));
	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, -0.1], N[(im * N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im * N[(re * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * im + im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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. lower-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. lower-fma.f64 69.3

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

   5. Applied rewrites 69.3%

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

   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. lower-*.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. lower-*.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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative 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. lower-*.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 31.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 3.9

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

   11. Applied rewrites 3.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({im}^{4} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {im}^{4}\right) \cdot re\right)} \]

       2. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {im}^{4}\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {im}^{4}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

       11. lower-*.f64 25.4

           \[\leadsto im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

   14. Applied rewrites 25.4%

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 55.6

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

   8. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied rewrites 57.3%

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

Alternative 7: 32.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right), \mathsf{fma}\left(re, im, im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.1)
     (fma
      im
      (fma im (* im (fma re -0.16666666666666666 -0.16666666666666666)) re)
      im)
     (if (<= t_0 0.0)
       (* im (* re (* 0.008333333333333333 (* (* im im) (* im im)))))
       (fma
        im
        (* (fma re 0.16666666666666666 0.5) (* re re))
        (fma re im im))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(im, fma(im, (im * fma(re, -0.16666666666666666, -0.16666666666666666)), re), im);
	} else if (t_0 <= 0.0) {
		tmp = im * (re * (0.008333333333333333 * ((im * im) * (im * im))));
	} else {
		tmp = fma(im, (fma(re, 0.16666666666666666, 0.5) * (re * re)), fma(re, im, im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = fma(im, fma(im, Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666)), re), im);
	elseif (t_0 <= 0.0)
		tmp = Float64(im * Float64(re * Float64(0.008333333333333333 * Float64(Float64(im * im) * Float64(im * im)))));
	else
		tmp = fma(im, Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)), fma(re, im, im));
	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, -0.1], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im * N[(re * N[(0.008333333333333333 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * im + im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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. lower-+.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 14.5%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. distribute-rgt-in N/A

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

       15. metadata-eval N/A

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

       16. lower-fma.f64 15.1

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

   11. Applied rewrites 15.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 3.9

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

   11. Applied rewrites 3.9%

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

   12. Taylor expanded in im around inf

       \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({im}^{4} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {im}^{4}\right) \cdot re\right)} \]

       2. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {im}^{4}\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{1}{120} \cdot {im}^{4}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)}\right) \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

       11. lower-*.f64 25.4

           \[\leadsto im \cdot \left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

   14. Applied rewrites 25.4%

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 55.6

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

   8. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied rewrites 57.3%

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

Alternative 8: 35.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(re, -0.16666666666666666, -0.16666666666666666\right), re\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-109)
   (fma
    im
    (fma im (* im (fma re -0.16666666666666666 -0.16666666666666666)) re)
    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)) <= 5e-109) {
		tmp = fma(im, fma(im, (im * fma(re, -0.16666666666666666, -0.16666666666666666)), re), 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)) <= 5e-109)
		tmp = fma(im, fma(im, Float64(im * fma(re, -0.16666666666666666, -0.16666666666666666)), re), 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], 5e-109], N[(im * N[(im * N[(im * N[(re * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] + im), $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)) < 5.0000000000000002e-109

   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. lower-+.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. distribute-rgt-in N/A

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

       15. metadata-eval N/A

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

       16. lower-fma.f64 36.9

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

   11. Applied rewrites 36.9%

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

   if 5.0000000000000002e-109 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 47.0

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

   8. Applied rewrites 47.0%

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

Alternative 9: 23.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* im (fma re (* -0.16666666666666666 (* im im)) re))
   (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(re * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 28.9%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 8.3

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

   11. Applied rewrites 8.3%

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

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 8.4

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

   14. Applied rewrites 8.4%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 57.3

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

   8. Applied rewrites 57.3%

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

Alternative 10: 23.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 28.9%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 8.3

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

   11. Applied rewrites 8.3%

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

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 8.4

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

   14. Applied rewrites 8.4%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 55.6

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

   8. Applied rewrites 55.6%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 57.3

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

   11. Applied rewrites 57.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 57.3

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

   13. Applied rewrites 57.3%

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

Alternative 11: 22.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 28.9%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 8.3

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

   11. Applied rewrites 8.3%

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

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 8.4

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

   14. Applied rewrites 8.4%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 55.6

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

   8. Applied rewrites 55.6%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 57.3

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

   11. Applied rewrites 57.3%

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

Alternative 12: 22.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* im (fma re (* -0.16666666666666666 (* im im)) re))
   (* im (fma re (fma re 0.5 1.0) 1.0))))

C

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

Julia

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

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 28.9%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 8.3

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

   11. Applied rewrites 8.3%

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

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 8.4

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

   14. Applied rewrites 8.4%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 56.4

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

   8. Applied rewrites 56.4%

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

Alternative 13: 34.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \left(im \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-6)
   (fma (* im im) (* im -0.16666666666666666) im)
   (* (* re re) (* re (* im 0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-6) {
		tmp = fma((im * im), (im * -0.16666666666666666), im);
	} else {
		tmp = (re * re) * (re * (im * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-6)
		tmp = fma(Float64(im * im), Float64(im * -0.16666666666666666), im);
	else
		tmp = Float64(Float64(re * re) * Float64(re * Float64(im * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(re * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 3.99999999999999982e-6

   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. lower-sin.f64 51.8

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

   5. Applied rewrites 51.8%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.2

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

   8. Applied rewrites 42.2%

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

   if 3.99999999999999982e-6 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.3%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 39.7

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 29.6

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

   11. Applied rewrites 29.6%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \left(im \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 33.8% accurate, 0.9× speedup

Math

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

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(Float64(im * im), Float64(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[(N[(im * im), $MachinePrecision] * N[(im * -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. lower-sin.f64 40.3

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

   5. Applied rewrites 40.3%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 28.3

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

   8. Applied rewrites 28.3%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 56.4

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

   8. Applied rewrites 56.4%

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

Alternative 15: 33.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-6)
   (fma (* im im) (* im -0.16666666666666666) im)
   (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-6) {
		tmp = fma((im * im), (im * -0.16666666666666666), im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-6)
		tmp = fma(Float64(im * im), Float64(im * -0.16666666666666666), im);
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 3.99999999999999982e-6

   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. lower-sin.f64 51.8

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

   5. Applied rewrites 51.8%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.2

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

   8. Applied rewrites 42.2%

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

   if 3.99999999999999982e-6 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.3%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 39.7

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 28.7

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

   11. Applied rewrites 28.7%

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

Alternative 16: 33.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.26:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.26) (fma im re im) (* im (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.26) {
		tmp = fma(im, re, im);
	} else {
		tmp = im * (re * (re * 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.26)
		tmp = fma(im, re, im);
	else
		tmp = Float64(im * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.26], N[(im * re + im), $MachinePrecision], N[(im * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.26000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 79.6

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

   5. Applied rewrites 79.6%

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

   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. lower-fma.f64 40.0

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

   8. Applied rewrites 40.0%

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

   if 0.26000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.2%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 30.2

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

   8. Applied rewrites 30.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 30.9

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

   11. Applied rewrites 30.9%

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

Alternative 17: 28.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.998:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.998) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.998) {
		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 ((exp(re) * sin(im)) <= 0.998d0) 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 ((Math.exp(re) * Math.sin(im)) <= 0.998) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.998:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.998)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.998)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.998], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.7%

      \[\leadsto im \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 34.7

         \[\leadsto \color{blue}{im} \]

   10. Applied rewrites 34.7%

       \[\leadsto \color{blue}{im} \]

   if 0.998 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 97.0%

      \[e^{re} \cdot \sin im \]
   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. lower-+.f64 10.1

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   5. Applied rewrites 10.1%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right) + im \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto im \cdot \left(re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right) + \color{blue}{im} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right), im\right)} \]

   8. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot \mathsf{fma}\left(im, re, im\right), \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), re\right), im\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right)}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto im \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \]

       4. associate-*l* N/A

          \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re\right) \]

       5. *-lft-identity N/A

          \[\leadsto im \cdot \left({im}^{2} \cdot \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \]

       7. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot re, re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot re, re\right) \]

       9. lower-*.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \]

       10. sub-neg N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \]

       11. *-commutative N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \left(\color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot re, re\right) \]

       12. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot re, re\right) \]

       13. associate-*l* N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \left(\color{blue}{im \cdot \left(im \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot re, re\right) \]

       14. metadata-eval N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{1}{120}, \frac{-1}{6}\right)} \cdot re, re\right) \]

       16. lower-*.f64 14.8

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot 0.008333333333333333}, -0.16666666666666666\right) \cdot re, re\right) \]

   11. Applied rewrites 14.8%

       \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot re} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot im} \]

       2. lower-*.f64 15.0

          \[\leadsto \color{blue}{re \cdot im} \]

   14. Applied rewrites 15.0%

       \[\leadsto \color{blue}{re \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 97.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.00028:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+95}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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))))
   (if (<= re -0.0022)
     (* (exp re) im)
     (if (<= re 0.00028)
       t_0
       (if (<= re 1.45e+95)
         (* (exp re) (fma im (* -0.16666666666666666 (* im im)) im))
         t_0)))))

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 tmp;
	if (re <= -0.0022) {
		tmp = exp(re) * im;
	} else if (re <= 0.00028) {
		tmp = t_0;
	} else if (re <= 1.45e+95) {
		tmp = exp(re) * fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = t_0;
	}
	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))
	tmp = 0.0
	if (re <= -0.0022)
		tmp = Float64(exp(re) * im);
	elseif (re <= 0.00028)
		tmp = t_0;
	elseif (re <= 1.45e+95)
		tmp = Float64(exp(re) * fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im));
	else
		tmp = t_0;
	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]}, If[LessEqual[re, -0.0022], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.00028], t$95$0, If[LessEqual[re, 1.45e+95], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00220000000000000013

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 100.0

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   if -0.00220000000000000013 < re < 2.7999999999999998e-4 or 1.45000000000000007e95 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \sin im \]

      2. lower-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. lower-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. lower-fma.f64 99.4

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \sin im \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]

   if 2.7999999999999998e-4 < re < 1.45000000000000007e95

   1. Initial program 95.7%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + e^{re}\right)} \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re}} + e^{re}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot e^{re}\right)} \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \cdot e^{re}\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot e^{re}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto e^{re} \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot 1\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto e^{re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto e^{re} \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{6} \cdot {im}^{2}, im\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \color{blue}{\frac{-1}{6} \cdot {im}^{2}}, im\right) \]

      14. unpow2 N/A

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

      15. lower-*.f64 82.1

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.00028:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+95}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.0% 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 99.6%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   2. lower-exp.f64 72.1

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Applied rewrites 72.1%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

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. lower-fma.f64 33.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

8. Applied rewrites 33.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]
9. Add Preprocessing

Alternative 20: 26.9% 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 99.6%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   2. lower-exp.f64 72.1

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Applied rewrites 72.1%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto im \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 30.6%

   \[\leadsto im \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 30.6

      \[\leadsto \color{blue}{im} \]

10. Applied rewrites 30.6%

    \[\leadsto \color{blue}{im} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))