math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 16.7s

Alternatives: 25

Speedup: 1.0×

• 24.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (* 0.16666666666666666 (* re (* re re)))
      (fma
       (fma
        (* im im)
        (fma (* im im) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* im (* im im))
       im))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 5e-103) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.16666666666666666 * (re * (re * re))) * fma(fma((im * im), fma((im * im), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (im * (im * im)), im);
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 5e-103) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.16666666666666666 * Float64(re * Float64(re * re))) * fma(fma(Float64(im * im), fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(im * Float64(im * im)), im));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 5e-103)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 5e-103], 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 im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 61.6

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

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

   11. Applied rewrites 65.7%

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

   12. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{re}^{3}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), im \cdot \left(im \cdot im\right), im\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 65.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999966e-103 < (*.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 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999966e-103 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. lower-*.f64 N/A

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

      2. lower-exp.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 61.6

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

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

   11. Applied rewrites 65.7%

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

   12. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{re}^{3}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), im \cdot \left(im \cdot im\right), im\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 65.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 63.5

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 52.9

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

   11. Applied rewrites 52.9%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 39.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 40.4

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

   8. Applied rewrites 40.4%

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

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

   11. Applied rewrites 43.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 49.2

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.3%

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

Alternative 5: 39.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot im - re \cdot \left(re \cdot \left(im \cdot im\right)\right)}{im - re \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.04)
     (*
      (* 0.16666666666666666 (* re (* re re)))
      (fma
       (fma
        (* im im)
        (fma (* im im) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* im (* im im))
       im))
     (if (<= t_0 0.0)
       (/ (- (* im im) (* re (* re (* im im)))) (- im (* re im)))
       (*
        (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
        (fma
         (* im im)
         (* im (fma (* im im) 0.008333333333333333 -0.16666666666666666))
         im))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = (0.16666666666666666 * (re * (re * re))) * fma(fma((im * im), fma((im * im), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (im * (im * im)), im);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) - (re * (re * (im * im)))) / (im - (re * im));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma((im * im), (im * fma((im * im), 0.008333333333333333, -0.16666666666666666)), im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(Float64(0.16666666666666666 * Float64(re * Float64(re * re))) * fma(fma(Float64(im * im), fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(im * Float64(im * im)), im));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) - Float64(re * Float64(re * Float64(im * im)))) / Float64(im - Float64(re * im)));
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.008333333333333333, -0.16666666666666666)), im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], N[(N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] - N[(re * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im - N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 40.4

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

   8. Applied rewrites 40.4%

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

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

   11. Applied rewrites 43.0%

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

   12. Taylor expanded in re around inf

       \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{re}^{3}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), im \cdot \left(im \cdot im\right), im\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 42.5%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 49.2

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 38.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 35.8

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

   8. Applied rewrites 35.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 49.2

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 38.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 34.4

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 34.5%

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

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(re, re \cdot \frac{1}{2}, re\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 35.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 49.2

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 40.4

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

   8. Applied rewrites 40.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 49.2

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.6%

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

Alternative 9: 38.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 40.4

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

   8. Applied rewrites 40.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 49.2

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

   8. Applied rewrites 49.2%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.6%

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

Alternative 10: 39.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 40.4

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

   8. Applied rewrites 40.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.3%

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

   10. Applied rewrites 36.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.4%

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

4. Final simplification 41.0%

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

Alternative 11: 38.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 33.6

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

   8. Applied rewrites 33.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.4%

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

Alternative 12: 26.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 33.6

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 15.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.4%

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

Alternative 13: 26.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 33.6

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 15.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.4%

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

4. Final simplification 26.0%

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

Alternative 14: 35.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 28.7

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

   8. Applied rewrites 28.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.4%

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

4. Final simplification 34.9%

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

Alternative 15: 35.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 28.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.4%

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

Alternative 16: 34.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-10)
   (fma im (* im (* im -0.16666666666666666)) im)
   (* im (fma (fma re 0.16666666666666666 0.5) (* re re) re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-10) {
		tmp = fma(im, (im * (im * -0.16666666666666666)), im);
	} else {
		tmp = im * fma(fma(re, 0.16666666666666666, 0.5), (re * re), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-10)
		tmp = fma(im, Float64(im * Float64(im * -0.16666666666666666)), im);
	else
		tmp = Float64(im * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-10], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000015e-10

   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 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 39.2%

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

   if 4.00000000000000015e-10 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 24.3

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

   5. Applied rewrites 24.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 17.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 17.6%

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

Alternative 17: 34.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-10)
   (fma im (* im (* im -0.16666666666666666)) im)
   (* im (* re (* re (fma re 0.16666666666666666 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-10) {
		tmp = fma(im, (im * (im * -0.16666666666666666)), im);
	} else {
		tmp = im * (re * (re * fma(re, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-10)
		tmp = fma(im, Float64(im * Float64(im * -0.16666666666666666)), im);
	else
		tmp = Float64(im * Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-10], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000015e-10

   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 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 39.2%

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

   if 4.00000000000000015e-10 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 24.3

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

   5. Applied rewrites 24.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 17.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 17.2%

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

Alternative 18: 34.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-10)
   (fma im (* im (* im -0.16666666666666666)) im)
   (* im (* re (* re (* re 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-10) {
		tmp = fma(im, (im * (im * -0.16666666666666666)), im);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-10)
		tmp = fma(im, Float64(im * Float64(im * -0.16666666666666666)), im);
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-10], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000015e-10

   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 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 39.2%

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

   if 4.00000000000000015e-10 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 24.3

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Applied rewrites 24.3%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 17.3%

      \[\leadsto im \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)}, 1\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot \left(\frac{1}{6} \cdot {re}^{\color{blue}{3}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 17.3%

       \[\leadsto im \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 33.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (fma im (* im (* im -0.16666666666666666)) im)
   (* im (fma re (fma re 0.5 1.0) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = fma(im, (im * (im * -0.16666666666666666)), im);
	} else {
		tmp = im * fma(re, fma(re, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = fma(im, Float64(im * Float64(im * -0.16666666666666666)), im);
	else
		tmp = Float64(im * fma(re, fma(re, 0.5, 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. lower-sin.f64 35.6

         \[\leadsto \color{blue}{\sin im} \]

   5. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.4%

      \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \left(im \cdot -0.16666666666666666\right)}, im\right) \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 51.9

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(1 + \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot re\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

      \[\leadsto im \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 33.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-10)
   (fma im (* im (* im -0.16666666666666666)) im)
   (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-10) {
		tmp = fma(im, (im * (im * -0.16666666666666666)), im);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-10)
		tmp = fma(im, Float64(im * Float64(im * -0.16666666666666666)), im);
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-10], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000015e-10

   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 45.3

         \[\leadsto \color{blue}{\sin im} \]

   5. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.2%

      \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \left(im \cdot -0.16666666666666666\right)}, im\right) \]

   if 4.00000000000000015e-10 < (*.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. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 24.3

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Applied rewrites 24.3%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   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 10.4%

      \[\leadsto \mathsf{fma}\left(im \cdot re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \left(im \cdot \color{blue}{{re}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.6%

       \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 31.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.96:\\ \;\;\;\;im \cdot 1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.96) (* im 1.0) (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.96) {
		tmp = im * 1.0;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 0.96d0) then
        tmp = im * 1.0d0
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.96) {
		tmp = im * 1.0;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.96:
		tmp = im * 1.0
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.96)
		tmp = Float64(im * 1.0);
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.96)
		tmp = im * 1.0;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.96], N[(im * 1.0), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.95999999999999996

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 69.6

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto im \cdot 1 \]

   if 0.95999999999999996 < (*.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. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 54.8

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   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 20.7%

      \[\leadsto \mathsf{fma}\left(im \cdot re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \left(im \cdot \color{blue}{{re}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 33.6%

       \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 39.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin im) 0.01)
   (*
    (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
    (fma im (* (* im im) -0.16666666666666666) im))
   (*
    im
    (fma
     (* im im)
     (fma (* im im) 0.008333333333333333 -0.16666666666666666)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(im) <= 0.01) {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, ((im * im) * -0.16666666666666666), im);
	} else {
		tmp = im * fma((im * im), fma((im * im), 0.008333333333333333, -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(im) <= 0.01)
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, Float64(Float64(im * im) * -0.16666666666666666), im));
	else
		tmp = Float64(im * fma(Float64(im * im), fma(Float64(im * im), 0.008333333333333333, -0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[im], $MachinePrecision], 0.01], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 0.0100000000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{re} \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto e^{re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto e^{re} \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{6} \cdot {im}^{2}, im\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \color{blue}{\frac{-1}{6} \cdot {im}^{2}}, im\right) \]

      6. unpow2 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

      7. lower-*.f64 78.9

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \color{blue}{\left(im \cdot im\right)}, im\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto e^{re} \cdot \color{blue}{\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, \frac{-1}{6} \cdot \left(im \cdot im\right), im\right) \]

      7. lower-fma.f64 49.6

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right) \]

   8. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right) \]

   if 0.0100000000000000002 < (sin.f64 im)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. lower-sin.f64 59.6

         \[\leadsto \color{blue}{\sin im} \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 8.5%

      \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 29.8% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(im, re, im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma im re im))

C

double code(double re, double im) {
	return fma(im, re, im);
}

Julia

function code(re, im)
	return fma(im, re, im)
end

Wolfram

code[re_, im_] := N[(im * re + im), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   2. lower-exp.f64 68.3

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Applied rewrites 68.3%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{im \cdot re} \]
7. Step-by-step derivation

8. Applied rewrites 27.7%

   \[\leadsto \mathsf{fma}\left(im, \color{blue}{re}, im\right) \]
9. Add Preprocessing

Alternative 24: 26.7% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ im \cdot 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im 1.0))

C

double code(double re, double im) {
	return im * 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * 1.0d0
end function

Java

public static double code(double re, double im) {
	return im * 1.0;
}

Python

def code(re, im):
	return im * 1.0

Julia

function code(re, im)
	return Float64(im * 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = im * 1.0;
end

Wolfram

code[re_, im_] := N[(im * 1.0), $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. lower-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   2. lower-exp.f64 68.3

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Applied rewrites 68.3%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto im \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 27.0%

   \[\leadsto im \cdot 1 \]
9. Add Preprocessing

Alternative 25: 6.9% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re im))

C

double code(double re, double im) {
	return re * im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * im
end function

Java

public static double code(double re, double im) {
	return re * im;
}

Python

def code(re, im):
	return re * im

Julia

function code(re, im)
	return Float64(re * im)
end

MATLAB

function tmp = code(re, im)
	tmp = re * im;
end

Wolfram

code[re_, im_] := N[(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. lower-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   2. lower-exp.f64 68.3

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Applied rewrites 68.3%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{im \cdot re} \]
7. Step-by-step derivation

8. Applied rewrites 27.7%

   \[\leadsto \mathsf{fma}\left(im, \color{blue}{re}, im\right) \]

9. Taylor expanded in re around inf

   \[\leadsto im \cdot re \]
10. Step-by-step derivation

11. Applied rewrites 4.5%

    \[\leadsto im \cdot re \]

12. Final simplification 4.5%

    \[\leadsto re \cdot im \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))