math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 20.6s

Alternatives: 20

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* im (* -0.16666666666666666 (* im im))))
     (if (<= t_0 -0.02)
       (* (sin im) (fma re (fma re 0.5 1.0) 1.0))
       (if (<= t_0 5e-68)
         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 = exp(re) * (im * (-0.16666666666666666 * (im * im)));
	} else if (t_0 <= -0.02) {
		tmp = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (t_0 <= 5e-68) {
		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(exp(re) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (t_0 <= -0.02)
		tmp = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_0 <= 5e-68)
		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[(N[Exp[re], $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-68], 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 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 72.0

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

   5. Simplified 72.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 28.0

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

   8. Simplified 28.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999971e-68 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 95.7%

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

   if 4.99999999999999971e-68 < (*.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. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-68}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), t\_0, \left(im \cdot im\right) \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 (- INFINITY))
     (*
      im
      (fma
       (fma -0.16666666666666666 (* im im) 1.0)
       t_0
       (*
        (* im im)
        (*
         t_0
         (*
          im
          (*
           im
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)))))))
     (if (<= t_1 -0.02)
       (* (sin im) (fma re (fma re 0.5 1.0) 1.0))
       (if (<= t_1 5e-68)
         t_2
         (if (<= t_1 1.0) (* (sin im) (+ re 1.0)) t_2))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = im * fma(fma(-0.16666666666666666, (im * im), 1.0), t_0, ((im * im) * (t_0 * (im * (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333))))));
	} else if (t_1 <= -0.02) {
		tmp = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (t_1 <= 5e-68) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(im * fma(fma(-0.16666666666666666, Float64(im * im), 1.0), t_0, Float64(Float64(im * im) * Float64(t_0 * Float64(im * Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)))))));
	elseif (t_1 <= -0.02)
		tmp = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_1 <= 5e-68)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 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[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-68], t$95$2, If[LessEqual[t$95$1, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{120} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 44.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)} \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999971e-68 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 95.7%

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

   if 4.99999999999999971e-68 < (*.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. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-68}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), t\_0, \left(im \cdot im\right) \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{\sin im}{\left(-re\right) - -1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 (- INFINITY))
     (*
      im
      (fma
       (fma -0.16666666666666666 (* im im) 1.0)
       t_0
       (*
        (* im im)
        (*
         t_0
         (*
          im
          (*
           im
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)))))))
     (if (<= t_1 -0.02)
       (/ (sin im) (- (- re) -1.0))
       (if (<= t_1 5e-68)
         t_2
         (if (<= t_1 1.0) (* (sin im) (+ re 1.0)) t_2))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = im * fma(fma(-0.16666666666666666, (im * im), 1.0), t_0, ((im * im) * (t_0 * (im * (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333))))));
	} else if (t_1 <= -0.02) {
		tmp = sin(im) / (-re - -1.0);
	} else if (t_1 <= 5e-68) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(im * fma(fma(-0.16666666666666666, Float64(im * im), 1.0), t_0, Float64(Float64(im * im) * Float64(t_0 * Float64(im * Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)))))));
	elseif (t_1 <= -0.02)
		tmp = Float64(sin(im) / Float64(Float64(-re) - -1.0));
	elseif (t_1 <= 5e-68)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 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[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[(N[Sin[im], $MachinePrecision] / N[((-re) - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-68], t$95$2, If[LessEqual[t$95$1, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{120} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 44.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)} \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.9

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

   5. Simplified 99.9%

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

      1. flip-+ N/A

         \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \sin im \]

      2. associate-*l/ N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

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

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

      13. metadata-eval 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in re around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin im}}{re + -1} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. sin-lowering-sin.f64 99.9

         \[\leadsto \frac{-\color{blue}{\sin im}}{re + -1} \]

   10. Simplified 99.9%

       \[\leadsto \frac{\color{blue}{-\sin im}}{re + -1} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999971e-68 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 95.7%

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

   if 4.99999999999999971e-68 < (*.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. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\frac{\sin im}{\left(-re\right) - -1}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-68}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \left(re + 1\right)\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_2 := e^{re} \cdot \sin im\\ t_3 := e^{re} \cdot im\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), t\_1, \left(im \cdot im\right) \cdot \left(t\_1 \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-68}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (+ re 1.0)))
        (t_1 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_2 (* (exp re) (sin im)))
        (t_3 (* (exp re) im)))
   (if (<= t_2 (- INFINITY))
     (*
      im
      (fma
       (fma -0.16666666666666666 (* im im) 1.0)
       t_1
       (*
        (* im im)
        (*
         t_1
         (*
          im
          (*
           im
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)))))))
     (if (<= t_2 -0.02)
       t_0
       (if (<= t_2 5e-68) t_3 (if (<= t_2 1.0) t_0 t_3))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * (re + 1.0);
	double t_1 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_2 = exp(re) * sin(im);
	double t_3 = exp(re) * im;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = im * fma(fma(-0.16666666666666666, (im * im), 1.0), t_1, ((im * im) * (t_1 * (im * (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333))))));
	} else if (t_2 <= -0.02) {
		tmp = t_0;
	} else if (t_2 <= 5e-68) {
		tmp = t_3;
	} else if (t_2 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * Float64(re + 1.0))
	t_1 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_2 = Float64(exp(re) * sin(im))
	t_3 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(im * fma(fma(-0.16666666666666666, Float64(im * im), 1.0), t_1, Float64(Float64(im * im) * Float64(t_1 * Float64(im * Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)))))));
	elseif (t_2 <= -0.02)
		tmp = t_0;
	elseif (t_2 <= 5e-68)
		tmp = t_3;
	elseif (t_2 <= 1.0)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1 + N[(N[(im * im), $MachinePrecision] * N[(t$95$1 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], t$95$0, If[LessEqual[t$95$2, 5e-68], t$95$3, If[LessEqual[t$95$2, 1.0], t$95$0, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{120} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 44.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999971e-68 < (*.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. +-lowering-+.f64 99.2

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

   5. Simplified 99.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999971e-68 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 95.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-68}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot im\\ t_2 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), t\_0, \left(im \cdot im\right) \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) im))
        (t_2 (* (exp re) (sin im))))
   (if (<= t_2 (- INFINITY))
     (*
      im
      (fma
       (fma -0.16666666666666666 (* im im) 1.0)
       t_0
       (*
        (* im im)
        (*
         t_0
         (*
          im
          (*
           im
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)))))))
     (if (<= t_2 -0.02)
       (sin im)
       (if (<= t_2 5e-68) t_1 (if (<= t_2 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * im;
	double t_2 = exp(re) * sin(im);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = im * fma(fma(-0.16666666666666666, (im * im), 1.0), t_0, ((im * im) * (t_0 * (im * (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333))))));
	} else if (t_2 <= -0.02) {
		tmp = sin(im);
	} else if (t_2 <= 5e-68) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * im)
	t_2 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(im * fma(fma(-0.16666666666666666, Float64(im * im), 1.0), t_0, Float64(Float64(im * im) * Float64(t_0 * Float64(im * Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)))))));
	elseif (t_2 <= -0.02)
		tmp = sin(im);
	elseif (t_2 <= 5e-68)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$2, 5e-68], t$95$1, If[LessEqual[t$95$2, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{120} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 44.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999971e-68 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 98.1

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

   5. Simplified 98.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999971e-68 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 95.7%

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

Alternative 7: 57.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), t\_0, \left(im \cdot im\right) \cdot \left(t\_0 \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (*
      im
      (fma
       (fma -0.16666666666666666 (* im im) 1.0)
       t_0
       (*
        (* im im)
        (*
         t_0
         (*
          im
          (*
           im
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)))))))
     (if (<= t_1 -0.02)
       (sin im)
       (if (<= t_1 0.0)
         (* (* im (* -0.16666666666666666 (* im im))) (+ re 1.0))
         (if (<= t_1 1.0) (sin im) (* im t_0)))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = im * fma(fma(-0.16666666666666666, (im * im), 1.0), t_0, ((im * im) * (t_0 * (im * (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333))))));
	} else if (t_1 <= -0.02) {
		tmp = sin(im);
	} else if (t_1 <= 0.0) {
		tmp = (im * (-0.16666666666666666 * (im * im))) * (re + 1.0);
	} else if (t_1 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(im * fma(fma(-0.16666666666666666, Float64(im * im), 1.0), t_0, Float64(Float64(im * im) * Float64(t_0 * Float64(im * Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)))))));
	elseif (t_1 <= -0.02)
		tmp = sin(im);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(im * Float64(-0.16666666666666666 * Float64(im * im))) * Float64(re + 1.0));
	elseif (t_1 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[im], $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{120} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 44.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 96.7

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

   5. Simplified 96.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 31.2

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

   8. Simplified 31.2%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 19.9

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

   11. Simplified 19.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 75.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 56.0

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

   8. Simplified 56.0%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(im \cdot im\right) \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + re), $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 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 59.4

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

   5. Simplified 59.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 25.0

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

   8. Simplified 25.0%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 17.4

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

   11. Simplified 17.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 86.8

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

   5. Simplified 86.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 55.1

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

   8. Simplified 55.1%

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

4. Final simplification 31.4%

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

Alternative 9: 31.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* im (* -0.16666666666666666 (* im im))) (+ re 1.0))
   (* 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 * (-0.16666666666666666 * (im * im))) * (re + 1.0);
	} else {
		tmp = im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $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 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 59.4

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

   5. Simplified 59.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 25.0

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

   8. Simplified 25.0%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 17.4

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

   11. Simplified 17.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 62.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 55.1

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

   8. Simplified 55.1%

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

4. Final simplification 31.4%

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

Alternative 10: 30.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(Float64(im * Float64(-0.16666666666666666 * Float64(im * im))) * Float64(re + 1.0));
	else
		tmp = fma(Float64(re * Float64(re * 0.5)), im, fma(re, 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[(N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] * im + N[(re * im + im), $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 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 59.4

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

   5. Simplified 59.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 25.0

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

   8. Simplified 25.0%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 17.4

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

   11. Simplified 17.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 62.1%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 50.1

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

   8. Simplified 50.1%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. accelerator-lowering-fma.f64 50.1

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

   10. Applied egg-rr 50.1%

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

4. Final simplification 29.5%

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

Alternative 11: 28.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(-0.16666666666666666 * Float64(im * Float64(im * fma(im, re, im))));
	else
		tmp = fma(Float64(re * Float64(re * 0.5)), im, fma(re, 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[(-0.16666666666666666 * N[(im * N[(im * N[(im * re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] * im + N[(re * im + im), $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 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 59.4

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

   5. Simplified 59.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 25.0

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

   8. Simplified 25.0%

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

   9. Taylor expanded in im around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

       9. +-commutative N/A

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

       10. distribute-lft-in N/A

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

       11. *-rgt-identity N/A

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

       12. accelerator-lowering-fma.f64 15.2

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

   11. Simplified 15.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 62.1%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 50.1

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

   8. Simplified 50.1%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. accelerator-lowering-fma.f64 50.1

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

   10. Applied egg-rr 50.1%

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

Alternative 12: 28.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.16666666666666666 * N[(im * N[(im * N[(im * re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision] + re), $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 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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 59.4

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

   5. Simplified 59.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 25.0

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

   8. Simplified 25.0%

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

   9. Taylor expanded in im around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

       9. +-commutative N/A

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

       10. distribute-lft-in N/A

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

       11. *-rgt-identity N/A

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

       12. accelerator-lowering-fma.f64 15.2

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

   11. Simplified 15.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 62.1%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 50.1

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

   8. Simplified 50.1%

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

Alternative 13: 34.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. sin-lowering-sin.f64 40.0

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

   5. Simplified 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 24.3

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

   8. Simplified 24.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 62.1%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 50.1

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

   8. Simplified 50.1%

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

Alternative 14: 33.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 1e-7)
   (fma im (* -0.16666666666666666 (* im im)) im)
   (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 1e-7) {
		tmp = fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 1e-7)
		tmp = fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im);
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 1e-7], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 50.0

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

   5. Simplified 50.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 37.2

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

   8. Simplified 37.2%

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

   if 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 39.1%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 21.1

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

   8. Simplified 21.1%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. accelerator-lowering-fma.f64 21.1

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

   10. Applied egg-rr 21.1%

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

   11. Taylor expanded in re around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 21.3

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

   13. Simplified 21.3%

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

Alternative 15: 33.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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 im around 0

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

   5. Simplified 71.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + im \cdot re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 33.8

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

   8. Simplified 33.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 75.9%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 39.3

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

   8. Simplified 39.3%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. accelerator-lowering-fma.f64 39.3

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

   10. Applied egg-rr 39.3%

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

   11. Taylor expanded in re around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 39.3

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

   13. Simplified 39.3%

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

Alternative 16: 28.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.999:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.999) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.999) {
		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)) <= 0.999d0) 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)) <= 0.999) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.999:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.999)
		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)) <= 0.999)
		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], 0.999], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 71.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 31.1%

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

   if 0.998999999999999999 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 73.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + im \cdot re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 12.4

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

   8. Simplified 12.4%

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

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 12.6

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

   11. Simplified 12.6%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.999:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;re \leq -0.078:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -0.078)
     (* (exp re) im)
     (if (<= re 9.6e-15)
       t_0
       (if (<= re 1e+103)
         (* (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 <= -0.078) {
		tmp = exp(re) * im;
	} else if (re <= 9.6e-15) {
		tmp = t_0;
	} else if (re <= 1e+103) {
		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 <= -0.078)
		tmp = Float64(exp(re) * im);
	elseif (re <= 9.6e-15)
		tmp = t_0;
	elseif (re <= 1e+103)
		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, -0.078], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 9.6e-15], t$95$0, If[LessEqual[re, 1e+103], 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 < -0.0779999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   if -0.0779999999999999999 < re < 9.5999999999999998e-15 or 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   if 9.5999999999999998e-15 < re < 1e103

   1. Initial program 99.8%

      \[e^{re} \cdot \sin im \]
   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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 92.4

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

   5. Simplified 92.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.078:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{if}\;re \leq -0.016:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (fma re (fma re 0.5 1.0) 1.0))))
   (if (<= re -0.016)
     (* (exp re) im)
     (if (<= re 9.6e-15)
       t_0
       (if (<= re 1.9e+154)
         (* (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, 0.5, 1.0), 1.0);
	double tmp;
	if (re <= -0.016) {
		tmp = exp(re) * im;
	} else if (re <= 9.6e-15) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		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, 0.5, 1.0), 1.0))
	tmp = 0.0
	if (re <= -0.016)
		tmp = Float64(exp(re) * im);
	elseif (re <= 9.6e-15)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		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 * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.016], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 9.6e-15], t$95$0, If[LessEqual[re, 1.9e+154], 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 < -0.016

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   if -0.016 < re < 9.5999999999999998e-15 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.8

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

   5. Simplified 99.8%

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

   if 9.5999999999999998e-15 < re < 1.8999999999999999e154

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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. *-lowering-*.f64 N/A

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

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

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

      13. *-lowering-*.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. *-lowering-*.f64 89.1

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

   5. Simplified 89.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.016:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.7% accurate, 29.4× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (fma im re im))

C

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

Julia

function code(re, im)
	return fma(im, re, im)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Simplified 71.9%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im + im \cdot re} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 31.5

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

8. Simplified 31.5%

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

Alternative 20: 26.6% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

double code(double re, double im) {
	return im;
}

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

function tmp = code(re, im)
	tmp = im;
end

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Simplified 71.9%

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

6. Taylor expanded in re around 0

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

8. Simplified 27.7%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))