math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 21

Speedup: 1.0×

• 25.6% 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: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.02 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-31} \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)) (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* im im) -0.16666666666666666 1.0) t_0)
     (if (or (<= t_1 -0.02) (not (or (<= t_1 5e-31) (not (<= t_1 1.0)))))
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
       t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * t_0;
	} else if ((t_1 <= -0.02) || !((t_1 <= 5e-31) || !(t_1 <= 1.0))) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * t_0);
	elseif ((t_1 <= -0.02) || !((t_1 <= 5e-31) || !(t_1 <= 1.0)))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.02], N[Not[Or[LessEqual[t$95$1, 5e-31], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh---cosh-rev N/A

         \[\leadsto \frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \cdot \sin im \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{e^{\mathsf{neg}\left(re\right)}}} \cdot \sin im \]

      6. sinh-cosh N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. exp-neg N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. exp-neg N/A

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

      11. associate-/r/ N/A

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

      12. /-rgt-identity N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

   10. Applied rewrites 82.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \left(e^{re} \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.02 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-31} \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)) (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* im im) -0.16666666666666666 1.0) t_0)
     (if (or (<= t_1 -0.02) (not (or (<= t_1 5e-31) (not (<= t_1 1.0)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * t_0;
	} else if ((t_1 <= -0.02) || !((t_1 <= 5e-31) || !(t_1 <= 1.0))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * t_0);
	elseif ((t_1 <= -0.02) || !((t_1 <= 5e-31) || !(t_1 <= 1.0)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.02], N[Not[Or[LessEqual[t$95$1, 5e-31], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh---cosh-rev N/A

         \[\leadsto \frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \cdot \sin im \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{e^{\mathsf{neg}\left(re\right)}}} \cdot \sin im \]

      6. sinh-cosh N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. exp-neg N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. exp-neg N/A

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

      11. associate-/r/ N/A

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

      12. /-rgt-identity N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

   10. Applied rewrites 82.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \left(e^{re} \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-31} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (pow im 3.0) -0.16666666666666666 im))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-31) (not (<= t_0 1.0)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma((im ^ 3.0), -0.16666666666666666, im));
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

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

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

      1. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 37.7

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 37.7%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-31} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (pow im 3.0) -0.16666666666666666 im))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-31) (not (<= t_0 1.0)))))
       (* (+ 1.0 re) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0))) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma((im ^ 3.0), -0.16666666666666666, im));
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0)))
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

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

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

      1. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 37.7

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 37.7%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.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. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-31} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-31) (not (<= t_0 1.0)))))
       (* (+ 1.0 re) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0))) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0)))
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

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 im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.5%

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

   10. Applied rewrites 43.5%

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

   12. Applied rewrites 32.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.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. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-31} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-31) (not (<= t_0 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-31) || !(t_0 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

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 im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.5%

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

   10. Applied rewrites 43.5%

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

   12. Applied rewrites 32.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - t\_1} \cdot im\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1 \cdot t\_1}{1 + -1 \cdot re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) re)))
   (if (<= t_0 (- INFINITY))
     (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 5e-31)
         (* (/ (fma (- re) re 1.0) (- 1.0 t_1)) im)
         (if (<= t_0 1.0)
           (sin im)
           (* (/ (- 1.0 (* t_1 t_1)) (+ 1.0 (* -1.0 re))) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 5e-31) {
		tmp = (fma(-re, re, 1.0) / (1.0 - t_1)) * im;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = ((1.0 - (t_1 * t_1)) / (1.0 + (-1.0 * re))) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 5e-31)
		tmp = Float64(Float64(fma(Float64(-re), re, 1.0) / Float64(1.0 - t_1)) * im);
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 + Float64(-1.0 * re))) * im);
	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[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 5e-31], N[(N[(N[((-re) * re + 1.0), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.5%

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

   10. Applied rewrites 43.5%

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

   12. Applied rewrites 32.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.0%

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

   10. Applied rewrites 47.5%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 56.9%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.8%

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

   10. Applied rewrites 19.8%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 67.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right)}{1 + -1 \cdot re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\\ \mathbf{if}\;t\_0 \leq -0.34:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - t\_1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1 \cdot t\_1}{1 + -1 \cdot re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) re)))
   (if (<= t_0 -0.34)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
     (if (<= t_0 0.0005)
       (* (/ (fma (- re) re 1.0) (- 1.0 t_1)) im)
       (* (/ (- 1.0 (* t_1 t_1)) (+ 1.0 (* -1.0 re))) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re;
	double tmp;
	if (t_0 <= -0.34) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else if (t_0 <= 0.0005) {
		tmp = (fma(-re, re, 1.0) / (1.0 - t_1)) * im;
	} else {
		tmp = ((1.0 - (t_1 * t_1)) / (1.0 + (-1.0 * re))) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re)
	tmp = 0.0
	if (t_0 <= -0.34)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	elseif (t_0 <= 0.0005)
		tmp = Float64(Float64(fma(Float64(-re), re, 1.0) / Float64(1.0 - t_1)) * im);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 + Float64(-1.0 * re))) * im);
	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[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[t$95$0, -0.34], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.0005], N[(N[(N[((-re) * re + 1.0), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.340000000000000024

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 24.0%

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

   10. Applied rewrites 24.0%

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

   12. Applied rewrites 18.4%

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

   if -0.340000000000000024 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   10. Applied rewrites 43.9%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 52.6%

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

   if 5.0000000000000001e-4 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.1%

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

   10. Applied rewrites 13.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 40.9%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.34:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right)}{1 + -1 \cdot re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right)\\ t_2 := t\_1 \cdot re\\ \mathbf{if}\;t\_0 \leq -0.34:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - t\_2} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{t\_2 \cdot t\_1}, \sqrt{re}, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma (fma 0.16666666666666666 re 0.5) re 1.0))
        (t_2 (* t_1 re)))
   (if (<= t_0 -0.34)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
     (if (<= t_0 1.0)
       (* (/ (fma (- re) re 1.0) (- 1.0 t_2)) im)
       (* (fma (sqrt (* t_2 t_1)) (sqrt re) 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(fma(0.16666666666666666, re, 0.5), re, 1.0);
	double t_2 = t_1 * re;
	double tmp;
	if (t_0 <= -0.34) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else if (t_0 <= 1.0) {
		tmp = (fma(-re, re, 1.0) / (1.0 - t_2)) * im;
	} else {
		tmp = fma(sqrt((t_2 * t_1)), sqrt(re), 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(fma(0.16666666666666666, re, 0.5), re, 1.0)
	t_2 = Float64(t_1 * re)
	tmp = 0.0
	if (t_0 <= -0.34)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(fma(Float64(-re), re, 1.0) / Float64(1.0 - t_2)) * im);
	else
		tmp = Float64(fma(sqrt(Float64(t_2 * t_1)), sqrt(re), 1.0) * im);
	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[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * re), $MachinePrecision]}, If[LessEqual[t$95$0, -0.34], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[((-re) * re + 1.0), $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[re], $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.340000000000000024

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 24.0%

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

   10. Applied rewrites 24.0%

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

   12. Applied rewrites 18.4%

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

   if -0.340000000000000024 < (*.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.2%

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

   10. Applied rewrites 37.8%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 45.1%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.8%

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

   10. Applied rewrites 53.8%

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

   12. Applied rewrites 59.1%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.34:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right)}, \sqrt{re}, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - t\_1 \cdot re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma (fma 0.16666666666666666 re 0.5) re 1.0)))
   (if (<= t_0 -0.3)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
     (if (<= t_0 0.0)
       (* (/ (fma (- re) re 1.0) (- 1.0 (* t_1 re))) im)
       (* (fma t_1 re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(fma(0.16666666666666666, re, 0.5), re, 1.0);
	double tmp;
	if (t_0 <= -0.3) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else if (t_0 <= 0.0) {
		tmp = (fma(-re, re, 1.0) / (1.0 - (t_1 * re))) * im;
	} else {
		tmp = fma(t_1, re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(fma(0.16666666666666666, re, 0.5), re, 1.0)
	tmp = 0.0
	if (t_0 <= -0.3)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(-re), re, 1.0) / Float64(1.0 - Float64(t_1 * re))) * im);
	else
		tmp = Float64(fma(t_1, re, 1.0) * im);
	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[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.3], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[((-re) * re + 1.0), $MachinePrecision] / N[(1.0 - N[(t$95$1 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(t$95$1 * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.299999999999999989

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 31.8

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.2%

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

   10. Applied rewrites 23.2%

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

   12. Applied rewrites 17.8%

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

   if -0.299999999999999989 < (*.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.2%

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

   10. Applied rewrites 31.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 42.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.9%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right)}{1 - \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.31:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), -re, -1\right), re, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.31)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) (- re) -1.0) re 1.0) im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.31) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), -re, -1.0), re, 1.0) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.31)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), Float64(-re), -1.0), re, 1.0) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.31], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * (-re) + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.309999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 32.4

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.6%

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

   10. Applied rewrites 23.6%

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

   12. Applied rewrites 18.1%

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

   if -0.309999999999999998 < (*.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 78.4

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.2%

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

Alternative 13: 37.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot im, 0.5, im\right), re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.1)
   (fma (fma (* re im) 0.5 im) re im)
   (* (* (fma 0.16666666666666666 re 0.5) (* re re)) im)))

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.1)
		tmp = fma(fma(Float64(re * im), 0.5, im), re, im);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, re, 0.5) * Float64(re * re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(N[(re * im), $MachinePrecision] * 0.5 + im), $MachinePrecision] * re + im), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.3%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.1%

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

Alternative 14: 36.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot im, 0.5, im\right), re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.1)
   (fma (fma (* re im) 0.5 im) re im)
   (* (* (* re re) im) 0.5)))

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.1)
		tmp = fma(fma(Float64(re * im), 0.5, im), re, im);
	else
		tmp = Float64(Float64(Float64(re * re) * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(N[(re * im), $MachinePrecision] * 0.5 + im), $MachinePrecision] * re + im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.3%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.5%

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

Alternative 15: 33.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.1:\\ \;\;\;\;\left(re - -1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.1)
   (* (- re -1.0) im)
   (* (* (* re re) im) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.1) {
		tmp = (re - -1.0) * im;
	} else {
		tmp = ((re * re) * im) * 0.5;
	}
	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)) <= 0.1d0) then
        tmp = (re - (-1.0d0)) * im
    else
        tmp = ((re * re) * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.1) {
		tmp = (re - -1.0) * im;
	} else {
		tmp = ((re * re) * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.1:
		tmp = (re - -1.0) * im
	else:
		tmp = ((re * re) * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.1)
		tmp = Float64(Float64(re - -1.0) * im);
	else
		tmp = Float64(Float64(Float64(re * re) * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.1)
		tmp = (re - -1.0) * im;
	else
		tmp = ((re * re) * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(re - -1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(1 + re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 33.7%

      \[\leadsto \left(re - -1\right) \cdot im \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.5%

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

Alternative 16: 39.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im))

C

double code(double re, double im) {
	return fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 37.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]
9. Add Preprocessing

Alternative 17: 39.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (* (* re re) 0.16666666666666666) re 1.0) im))

C

double code(double re, double im) {
	return fma(((re * re) * 0.16666666666666666), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 37.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 37.1%

    \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im \]
12. Add Preprocessing

Alternative 18: 37.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (fma (fma 0.5 re 1.0) re 1.0) im))

C

double code(double re, double im) {
	return fma(fma(0.5, re, 1.0), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 34.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \]
9. Add Preprocessing

Alternative 19: 29.6% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \left(re - -1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- re -1.0) im))

C

double code(double re, double im) {
	return (re - -1.0) * im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (re - (-1.0d0)) * im
end function

Java

public static double code(double re, double im) {
	return (re - -1.0) * im;
}

Python

def code(re, im):
	return (re - -1.0) * im

Julia

function code(re, im)
	return Float64(Float64(re - -1.0) * im)
end

MATLAB

function tmp = code(re, im)
	tmp = (re - -1.0) * im;
end

Wolfram

code[re_, im_] := N[(N[(re - -1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in re around 0

   \[\leadsto \left(1 + re\right) \cdot im \]
7. Step-by-step derivation

8. Applied rewrites 28.7%

   \[\leadsto \left(re - -1\right) \cdot im \]
9. Add Preprocessing

Alternative 20: 29.6% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, im, im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma re im im))

C

double code(double re, double im) {
	return fma(re, im, im);
}

Julia

function code(re, im)
	return fma(re, im, im)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.7%

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

Alternative 21: 7.0% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return im * re;
}

Python

def code(re, im):
	return im * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.7%

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

9. Taylor expanded in re around inf

   \[\leadsto im \cdot re \]
10. Step-by-step derivation

11. Applied rewrites 6.7%

    \[\leadsto im \cdot re \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))