math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 16.9s

Alternatives: 25

Speedup: 1.0×

• 24.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 92.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right) \cdot e^{re}\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* (* im im) -0.16666666666666666) im im) (exp re))
     (if (<= t_0 -0.02)
       (* (+ (fma (* re re) 0.5 1.0) re) (sin im))
       (if (<= t_0 1e-80) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im) * exp(re);
	} else if (t_0 <= -0.02) {
		tmp = (fma((re * re), 0.5, 1.0) + re) * sin(im);
	} else if (t_0 <= 1e-80) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(im * im) * -0.16666666666666666), im, im) * exp(re));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, 1.0) + re) * sin(im));
	elseif (t_0 <= 1e-80)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-80], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

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

   11. Taylor expanded in re around inf

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

       1. lower-exp.f64 82.1

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

   13. Applied rewrites 82.1%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 96.9

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 96.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999961e-81 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 92.4

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

   5. Applied rewrites 92.4%

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

   if 9.99999999999999961e-81 < (*.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 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right) \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-80}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma 0.16666666666666666 re 0.5) (* re re))
      (fma (* (* im im) -0.16666666666666666) im im))
     (if (<= t_0 -0.02)
       (* (+ (fma (* re re) 0.5 1.0) re) (sin im))
       (if (<= t_0 1e-80) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(0.16666666666666666, re, 0.5) * (re * re)) * fma(((im * im) * -0.16666666666666666), im, im);
	} else if (t_0 <= -0.02) {
		tmp = (fma((re * re), 0.5, 1.0) + re) * sin(im);
	} else if (t_0 <= 1e-80) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.16666666666666666, re, 0.5) * Float64(re * re)) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, 1.0) + re) * sin(im));
	elseif (t_0 <= 1e-80)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-80], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 54.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 96.9

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 96.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999961e-81 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 92.4

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

   5. Applied rewrites 92.4%

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

   if 9.99999999999999961e-81 < (*.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 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-80}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma 0.16666666666666666 re 0.5) (* re re))
      (fma (* (* im im) -0.16666666666666666) im im))
     (if (<= t_0 -0.02)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (if (<= t_0 1e-80) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(0.16666666666666666, re, 0.5) * (re * re)) * fma(((im * im) * -0.16666666666666666), im, im);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (t_0 <= 1e-80) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.16666666666666666, re, 0.5) * Float64(re * re)) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	elseif (t_0 <= -0.02)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (t_0 <= 1e-80)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-80], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 54.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 96.9

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

   5. Applied rewrites 96.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999961e-81 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 92.4

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

   5. Applied rewrites 92.4%

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

   if 9.99999999999999961e-81 < (*.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 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-80}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma 0.16666666666666666 re 0.5) (* re re))
      (fma (* (* im im) -0.16666666666666666) im im))
     (if (<= t_0 -0.02)
       (* (+ 1.0 re) (sin im))
       (if (<= t_0 1e-80) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(0.16666666666666666, re, 0.5) * (re * re)) * fma(((im * im) * -0.16666666666666666), im, im);
	} else if (t_0 <= -0.02) {
		tmp = (1.0 + re) * sin(im);
	} else if (t_0 <= 1e-80) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.16666666666666666, re, 0.5) * Float64(re * re)) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(1.0 + re) * sin(im));
	elseif (t_0 <= 1e-80)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-80], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 54.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999961e-81 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 92.4

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

   5. Applied rewrites 92.4%

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

   if 9.99999999999999961e-81 < (*.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 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-80}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma 0.16666666666666666 re 0.5) (* re re))
      (fma (* (* im im) -0.16666666666666666) im im))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 1e-80) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(0.16666666666666666, re, 0.5) * (re * re)) * fma(((im * im) * -0.16666666666666666), im, im);
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 1e-80) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.16666666666666666, re, 0.5) * Float64(re * re)) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 1e-80)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-80], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 54.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 9.99999999999999961e-81 < (*.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 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999961e-81 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 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-80}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{0.16666666666666666 - \frac{0.6666666666666666}{re}} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (fma 0.16666666666666666 re 0.5) (* re re))
      (fma (* (* im im) -0.16666666666666666) im im))
     (if (<= t_0 1.0)
       (sin im)
       (*
        (/
         (fma
          (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
          (/ 1.0 (+ 1.0 re))
          (fma re 0.16666666666666666 -0.5))
         (- 0.16666666666666666 (/ 0.6666666666666666 re)))
        im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(0.16666666666666666, re, 0.5) * (re * re)) * fma(((im * im) * -0.16666666666666666), im, im);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = (fma((fma(0.027777777777777776, (re * re), -0.25) * (re * re)), (1.0 / (1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / (0.16666666666666666 - (0.6666666666666666 / re))) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.16666666666666666, re, 0.5) * Float64(re * re)) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)), Float64(1.0 / Float64(1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / Float64(0.16666666666666666 - Float64(0.6666666666666666 / re))) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + re), $MachinePrecision]), $MachinePrecision] + N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - N[(0.6666666666666666 / re), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 54.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 66.2

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

   5. Applied rewrites 66.2%

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

   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 65.8

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

   5. Applied rewrites 65.8%

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

   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 62.9%

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

   10. Applied rewrites 65.9%

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

   11. Taylor expanded in re around inf

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{36}, re \cdot re, \frac{-1}{4}\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot 1\right)}{\frac{1}{6} - \frac{2}{3} \cdot \frac{1}{re}} \cdot im \]
   12. Step-by-step derivation

   13. Applied rewrites 65.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 1\right)}{0.16666666666666666 - \frac{0.6666666666666666}{re}} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{0.16666666666666666 - \frac{0.6666666666666666}{re}} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 31.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 31.8%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 29.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   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 56.7%

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

   10. Applied rewrites 57.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 1\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \frac{1}{1 + re}} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

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

Alternative 9: 38.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + re}\\ \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), t\_0, 0.16666666666666666 \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot t\_0} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 re))))
   (if (<= (* (sin im) (exp re)) 5e-8)
     (*
      (fma (fma 0.5 re 1.0) re 1.0)
      (fma (* (* im im) -0.16666666666666666) im im))
     (*
      (/
       (fma
        (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
        t_0
        (* 0.16666666666666666 re))
       (* (fma re 0.16666666666666666 -0.5) t_0))
      im))))

C

double code(double re, double im) {
	double t_0 = 1.0 / (1.0 + re);
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-8) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = (fma((fma(0.027777777777777776, (re * re), -0.25) * (re * re)), t_0, (0.16666666666666666 * re)) / (fma(re, 0.16666666666666666, -0.5) * t_0)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(1.0 / Float64(1.0 + re))
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-8)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	else
		tmp = Float64(Float64(fma(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)), t_0, Float64(0.16666666666666666 * re)) / Float64(fma(re, 0.16666666666666666, -0.5) * t_0)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision] / N[(N[(re * 0.16666666666666666 + -0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.9999999999999998e-8

   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 44.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 42.6%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   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 34.1

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

   5. Applied rewrites 34.1%

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

   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 32.7%

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

   10. Applied rewrites 34.2%

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

   11. Taylor expanded in re around inf

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{36}, re \cdot re, \frac{-1}{4}\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \frac{1}{6} \cdot re\right)}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot \frac{1}{1 + re}} \cdot im \]
   12. Step-by-step derivation

   13. Applied rewrites 34.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, 0.16666666666666666 \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \frac{1}{1 + re}} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, 0.16666666666666666 \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \frac{1}{1 + re}} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{0.16666666666666666 - \frac{0.6666666666666666}{re}} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-8)
   (*
    (fma (fma 0.5 re 1.0) re 1.0)
    (fma (* (* im im) -0.16666666666666666) im im))
   (*
    (/
     (fma
      (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
      (/ 1.0 (+ 1.0 re))
      (fma re 0.16666666666666666 -0.5))
     (- 0.16666666666666666 (/ 0.6666666666666666 re)))
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-8) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = (fma((fma(0.027777777777777776, (re * re), -0.25) * (re * re)), (1.0 / (1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / (0.16666666666666666 - (0.6666666666666666 / re))) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-8)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	else
		tmp = Float64(Float64(fma(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)), Float64(1.0 / Float64(1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / Float64(0.16666666666666666 - Float64(0.6666666666666666 / re))) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + re), $MachinePrecision]), $MachinePrecision] + N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - N[(0.6666666666666666 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.9999999999999998e-8

   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 44.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 42.6%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   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 34.1

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

   5. Applied rewrites 34.1%

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

   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 32.7%

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

   10. Applied rewrites 34.2%

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

   11. Taylor expanded in re around inf

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{36}, re \cdot re, \frac{-1}{4}\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot 1\right)}{\frac{1}{6} - \frac{2}{3} \cdot \frac{1}{re}} \cdot im \]
   12. Step-by-step derivation

   13. Applied rewrites 34.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 1\right)}{0.16666666666666666 - \frac{0.6666666666666666}{re}} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{0.16666666666666666 - \frac{0.6666666666666666}{re}} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{0.16666666666666666} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-8)
   (*
    (fma (fma 0.5 re 1.0) re 1.0)
    (fma (* (* im im) -0.16666666666666666) im im))
   (*
    (/
     (fma
      (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
      (/ 1.0 (+ 1.0 re))
      (fma re 0.16666666666666666 -0.5))
     0.16666666666666666)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-8) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = (fma((fma(0.027777777777777776, (re * re), -0.25) * (re * re)), (1.0 / (1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / 0.16666666666666666) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-8)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	else
		tmp = Float64(Float64(fma(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)), Float64(1.0 / Float64(1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / 0.16666666666666666) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + re), $MachinePrecision]), $MachinePrecision] + N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] / 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.9999999999999998e-8

   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 44.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 42.6%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   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 34.1

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

   5. Applied rewrites 34.1%

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

   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 32.7%

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

   10. Applied rewrites 34.2%

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

   11. Taylor expanded in re around inf

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{36}, re \cdot re, \frac{-1}{4}\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot 1\right)}{\frac{1}{6}} \cdot im \]
   12. Step-by-step derivation

   13. Applied rewrites 33.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 1\right)}{0.16666666666666666} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{0.16666666666666666} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 31.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 31.8%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 29.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   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 56.7%

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

   10. Applied rewrites 56.7%

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

4. Final simplification 40.1%

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

Alternative 13: 37.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, \frac{t\_0}{-re}\right) \cdot \left(-re\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 re 1.0) re 1.0)))
   (if (<= (* (sin im) (exp re)) 5e-8)
     (* t_0 (fma (* (* im im) -0.16666666666666666) im im))
     (* (* (fma (* re re) -0.16666666666666666 (/ t_0 (- re))) (- re)) im))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.5, re, 1.0), re, 1.0);
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-8) {
		tmp = t_0 * fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = (fma((re * re), -0.16666666666666666, (t_0 / -re)) * -re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(0.5, re, 1.0), re, 1.0)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-8)
		tmp = Float64(t_0 * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.16666666666666666, Float64(t_0 / Float64(-re))) * Float64(-re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(t$95$0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + N[(t$95$0 / (-re)), $MachinePrecision]), $MachinePrecision] * (-re)), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.9999999999999998e-8

   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 44.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 42.6%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   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 34.1

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

   5. Applied rewrites 34.1%

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

   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 32.7%

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

   10. Applied rewrites 34.2%

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

   11. Taylor expanded in re around -inf

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

   13. Applied rewrites 32.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)}{-re}\right) \cdot \left(-re\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-8)
   (*
    (fma (fma 0.5 re 1.0) re 1.0)
    (fma (* (* im im) -0.16666666666666666) im 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 ((sin(im) * exp(re)) <= 5e-8) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma(((im * im) * -0.16666666666666666), im, 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(sin(im) * exp(re)) <= 5e-8)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, 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[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $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)) < 4.9999999999999998e-8

   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 44.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 42.6%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   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 34.1

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

   5. Applied rewrites 34.1%

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

   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 32.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.0)
   (* (- re -1.0) (fma (* (* im im) -0.16666666666666666) im 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 ((sin(im) * exp(re)) <= 0.0) {
		tmp = (re - -1.0) * fma(((im * im) * -0.16666666666666666), im, 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(sin(im) * exp(re)) <= 0.0)
		tmp = Float64(Float64(re - -1.0) * fma(Float64(Float64(im * im) * -0.16666666666666666), im, 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[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(re - -1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $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)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 31.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 31.8%

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

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. sub-neg N/A

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

       4. lower--.f64 22.7

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

   13. Applied rewrites 22.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   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 56.7%

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

4. Final simplification 36.1%

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

Alternative 16: 34.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.0)
   (* 1.0 (fma (* (* im im) -0.16666666666666666) im 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 ((sin(im) * exp(re)) <= 0.0) {
		tmp = 1.0 * fma(((im * im) * -0.16666666666666666), im, 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(sin(im) * exp(re)) <= 0.0)
		tmp = Float64(1.0 * fma(Float64(Float64(im * im) * -0.16666666666666666), im, 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[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $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)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 31.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 31.8%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 20.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   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 56.7%

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

4. Final simplification 35.0%

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

Alternative 17: 33.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-8)
   (* 1.0 (fma (* (* im im) -0.16666666666666666) im im))
   (* (fma (* (* re re) 0.16666666666666666) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-8) {
		tmp = 1.0 * fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-8)
		tmp = Float64(1.0 * fma(Float64(Float64(im * im) * -0.16666666666666666), im, im));
	else
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.9999999999999998e-8

   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 44.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 35.7%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   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 34.1

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

   5. Applied rewrites 34.1%

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

   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 32.7%

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

   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 32.7%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.35:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.35)
   (* 1.0 im)
   (* (* (* (fma 0.16666666666666666 re 0.5) re) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.35) {
		tmp = 1.0 * im;
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.35)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.35], N[(1.0 * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.34999999999999998

   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

      \[\leadsto 1 \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 33.1%

      \[\leadsto 1 \cdot im \]

   if 0.34999999999999998 < (*.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 39.0

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

   5. Applied rewrites 39.0%

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

   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 37.3%

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

   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 37.7%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.35:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 132000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right) \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75)
   (* im (exp re))
   (if (<= re 132000000.0)
     (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
     (if (<= re 1.02e+103)
       (* (fma (* (* im im) -0.16666666666666666) im im) (exp re))
       (* (fma (* (* re re) 0.16666666666666666) re 1.0) (sin im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75) {
		tmp = im * exp(re);
	} else if (re <= 132000000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (re <= 1.02e+103) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im) * exp(re);
	} else {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.75)
		tmp = Float64(im * exp(re));
	elseif (re <= 132000000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (re <= 1.02e+103)
		tmp = Float64(fma(Float64(Float64(im * im) * -0.16666666666666666), im, im) * exp(re));
	else
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.75], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 132000000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.75

   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%

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

   if -1.75 < re < 1.32e8

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 97.7

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

   5. Applied rewrites 97.7%

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

   if 1.32e8 < re < 1.01999999999999991e103

   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 4.0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 25.2

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 25.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.2%

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

   11. Taylor expanded in re around inf

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

       1. lower-exp.f64 94.4

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

   13. Applied rewrites 94.4%

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

   if 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 132000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right) \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.0% 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 69.9

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

5. Applied rewrites 69.9%

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

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 41.8%

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

9. 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 41.5%

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

Alternative 21: 35.9% 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 69.9

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

5. Applied rewrites 69.9%

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

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 37.7%

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

Alternative 22: 33.1% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
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 69.9

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

5. Applied rewrites 69.9%

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

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 34.8%

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

9. Final simplification 34.8%

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

Alternative 23: 27.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.15 \cdot 10^{+56}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3.15e+56) (* 1.0 im) (* im re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.15e+56) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.15d+56) then
        tmp = 1.0d0 * im
    else
        tmp = im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.15e+56) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.15e+56:
		tmp = 1.0 * im
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.15e+56)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(im * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.15e+56)
		tmp = 1.0 * im;
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.15e+56], N[(1.0 * im), $MachinePrecision], N[(im * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.15e56

   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 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 33.1%

      \[\leadsto 1 \cdot im \]

   if 3.15e56 < 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 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 8.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 8.9%

       \[\leadsto re \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.15 \cdot 10^{+56}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.1% 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 69.9

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

5. Applied rewrites 69.9%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.2%

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

Alternative 25: 6.8% 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 69.9

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

5. Applied rewrites 69.9%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.2%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 6.6%

    \[\leadsto re \cdot im \]

12. Final simplification 6.6%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))