math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 21.7s

Alternatives: 21

Speedup: 1.0×

• 24.7% 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: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (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 2e-92) 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 = fma(re, fma(re, 0.5, 1.0), 1.0) * 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 <= 2e-92) {
		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(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, Float64(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 <= 2e-92)
		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[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $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, 2e-92], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 47.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 1.99999999999999998e-92 < (*.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. Simplified 100.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.99999999999999998e-92 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 94.8

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

   5. Simplified 94.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-92}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, im \cdot 0.008333333333333333, -0.16666666666666666\right)\\ t_1 := im \cdot \left(im \cdot im\right)\\ t_2 := e^{re} \cdot \sin im\\ t_3 := \mathsf{fma}\left(t\_1, t\_0, -im\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), t\_1, im\right)\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, im\right) \cdot t\_3}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma im (* im 0.008333333333333333) -0.16666666666666666))
        (t_1 (* im (* im im)))
        (t_2 (* (exp re) (sin im)))
        (t_3 (fma t_1 t_0 (- im))))
   (if (<= t_2 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (fma
        im
        (* im (fma (* im im) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       t_1
       im))
     (if (<= t_2 -0.02)
       (sin im)
       (if (<= t_2 0.0)
         (/ (* (fma t_1 t_0 im) t_3) t_3)
         (if (<= t_2 1.0)
           (sin im)
           (*
            im
            (fma
             re
             (fma
              re
              (/ (fma (* re (* re re)) 0.004629629629629629 0.125) 0.25)
              1.0)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = fma(im, (im * 0.008333333333333333), -0.16666666666666666);
	double t_1 = im * (im * im);
	double t_2 = exp(re) * sin(im);
	double t_3 = fma(t_1, t_0, -im);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), t_1, im);
	} else if (t_2 <= -0.02) {
		tmp = sin(im);
	} else if (t_2 <= 0.0) {
		tmp = (fma(t_1, t_0, im) * t_3) / t_3;
	} else if (t_2 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = im * fma(re, fma(re, (fma((re * (re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, Float64(im * 0.008333333333333333), -0.16666666666666666)
	t_1 = Float64(im * Float64(im * im))
	t_2 = Float64(exp(re) * sin(im))
	t_3 = fma(t_1, t_0, Float64(-im))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), t_1, im));
	elseif (t_2 <= -0.02)
		tmp = sin(im);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(t_1, t_0, im) * t_3) / t_3);
	elseif (t_2 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(im * fma(re, fma(re, Float64(fma(Float64(re * Float64(re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$0 + (-im)), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * t$95$1 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(t$95$1 * t$95$0 + im), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[Sin[im], $MachinePrecision], N[(im * N[(re * N[(re * N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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 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 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 47.6%

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

   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 99.2

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

   5. Simplified 99.2%

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

   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 re around 0

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

      1. lower-sin.f64 37.8

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

   5. Simplified 37.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.4

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

   8. Simplified 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), im\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

   10. Applied egg-rr 32.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 75.8

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

   5. Simplified 75.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 55.4

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

   8. Simplified 55.4%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-fma.f64 N/A

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

      5. cube-mult N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 19.0

          \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot \color{blue}{0.08333333333333333}}, 1\right), 1\right) \]

   10. Applied egg-rr 19.0%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in re around 0

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

   13. Simplified 58.3%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (fma
        im
        (* im (fma (* im im) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       (* im (* im im))
       im))
     (if (<= t_0 1.0)
       (sin im)
       (*
        im
        (fma
         re
         (fma
          re
          (/ (fma (* re (* re re)) 0.004629629629629629 0.125) 0.25)
          1.0)
         1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (im * (im * im)), im);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = im * fma(re, fma(re, (fma((re * (re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(im * Float64(im * im)), im));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(im * fma(re, fma(re, Float64(fma(Float64(re * Float64(re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 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, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(im * N[(re * N[(re * N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 47.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 66.8

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

   5. Simplified 66.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 75.8

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

   5. Simplified 75.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 55.4

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

   8. Simplified 55.4%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-fma.f64 N/A

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

      5. cube-mult N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 19.0

          \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot \color{blue}{0.08333333333333333}}, 1\right), 1\right) \]

   10. Applied egg-rr 19.0%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in re around 0

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

   13. Simplified 58.3%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 35.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \mathsf{fma}\left(0.5, re \cdot re, re\right)\\ t_2 := im \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot im - t\_1 \cdot \left(im \cdot t\_2\right)}{im - t\_2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (fma 0.5 (* re re) re))
        (t_2 (* im t_1)))
   (if (<= t_0 -0.02)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (fma
        im
        (* im (fma (* im im) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       (* im (* im im))
       im))
     (if (<= t_0 0.0)
       (/ (- (* im im) (* t_1 (* im t_2))) (- im t_2))
       (*
        im
        (fma
         re
         (fma
          re
          (/ (fma (* re (* re re)) 0.004629629629629629 0.125) 0.25)
          1.0)
         1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = fma(0.5, (re * re), re);
	double t_2 = im * t_1;
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * 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) - (t_1 * (im * t_2))) / (im - t_2);
	} else {
		tmp = im * fma(re, fma(re, (fma((re * (re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = fma(0.5, Float64(re * re), re)
	t_2 = Float64(im * t_1)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, Float64(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(t_1 * Float64(im * t_2))) / Float64(im - t_2));
	else
		tmp = Float64(im * fma(re, fma(re, Float64(fma(Float64(re * Float64(re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0));
	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[(0.5 * N[(re * re), $MachinePrecision] + re), $MachinePrecision]}, Block[{t$95$2 = N[(im * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $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[(t$95$1 * N[(im * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im - t$95$2), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 25.5%

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

   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. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 37.0

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

   8. Simplified 37.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

   10. Applied egg-rr 21.2%

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

   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 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-fma.f64 N/A

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

      5. cube-mult N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 35.5

          \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot \color{blue}{0.08333333333333333}}, 1\right), 1\right) \]

   10. Applied egg-rr 35.5%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in re around 0

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

   13. Simplified 49.0%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 40.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-5)
   (*
    (fma re (fma re 0.5 1.0) 1.0)
    (fma
     (fma
      im
      (* im (fma (* im im) -0.0001984126984126984 0.008333333333333333))
      -0.16666666666666666)
     (* im (* im im))
     im))
   (*
    im
    (fma
     re
     (fma re (/ (fma (* re (* re re)) 0.004629629629629629 0.125) 0.25) 1.0)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 5e-5) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (im * (im * im)), im);
	} else {
		tmp = im * fma(re, fma(re, (fma((re * (re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-5)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(im, Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(im * Float64(im * im)), im));
	else
		tmp = Float64(im * fma(re, fma(re, Float64(fma(Float64(re * Float64(re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.00000000000000024e-5

   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 56.4

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

   5. Simplified 56.4%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 42.5%

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

   if 5.00000000000000024e-5 < (*.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 37.9

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

   5. Simplified 37.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 28.2

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

   8. Simplified 28.2%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-fma.f64 N/A

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

      5. cube-mult N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 10.8

          \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot \color{blue}{0.08333333333333333}}, 1\right), 1\right) \]

   10. Applied egg-rr 10.8%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in re around 0

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

   13. Simplified 29.6%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 37.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (fma
    (fma
     im
     (* im (fma (* im im) -0.0001984126984126984 0.008333333333333333))
     -0.16666666666666666)
    (* im (* im im))
    im)
   (*
    im
    (fma
     re
     (fma re (/ (fma (* re (* re re)) 0.004629629629629629 0.125) 0.25) 1.0)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = fma(fma(im, (im * fma((im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (im * (im * im)), im);
	} else {
		tmp = im * fma(re, fma(re, (fma((re * (re * re)), 0.004629629629629629, 0.125) / 0.25), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = fma(fma(im, Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(im * Float64(im * im)), im);
	else
		tmp = Float64(im * fma(re, fma(re, Float64(fma(Float64(re * Float64(re * re)), 0.004629629629629629, 0.125) / 0.25), 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[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $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 42.1

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

   5. Simplified 42.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 28.8%

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

   if 0.0 < (*.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 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-fma.f64 N/A

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

      5. cube-mult N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 35.5

          \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot \color{blue}{0.08333333333333333}}, 1\right), 1\right) \]

   10. Applied egg-rr 35.5%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, 0.25\right) - re \cdot 0.08333333333333333}}, 1\right), 1\right) \]

   11. Taylor expanded in re around 0

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

   13. Simplified 49.0%

       \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = fma(fma(im, Float64(im * fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(im * Float64(im * im)), im);
	else
		tmp = Float64(im * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), Float64(re + 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[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $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 42.1

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

   5. Simplified 42.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 28.8%

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

   if 0.0 < (*.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 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 48.0

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

   10. Applied egg-rr 48.0%

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

Alternative 9: 35.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right), im \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (fma (* im (* im (* im (* im -0.0001984126984126984)))) (* im (* im im)) im)
   (* im (fma (fma re 0.16666666666666666 0.5) (* re re) (+ re 1.0)))))

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = fma(Float64(im * Float64(im * Float64(im * Float64(im * -0.0001984126984126984)))), Float64(im * Float64(im * im)), im);
	else
		tmp = Float64(im * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), Float64(re + 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 * N[(im * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $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 42.1

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

   5. Simplified 42.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{im}^{3}} + im \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, {im}^{3}, im\right)} \]

   8. Simplified 28.8%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, im \cdot \left(im \cdot im\right), im\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   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 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 48.0

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

   10. Applied egg-rr 48.0%

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

Alternative 10: 35.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.9

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

   5. Simplified 41.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 28.6

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

   8. Simplified 28.6%

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

   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 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 48.0

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

   10. Applied egg-rr 48.0%

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

Alternative 11: 35.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.9

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

   5. Simplified 41.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 28.6

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

   8. Simplified 28.6%

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

   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 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

Alternative 12: 35.4% accurate, 0.9× speedup

Math

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

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(Float64(im * 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[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $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 42.1

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

   5. Simplified 42.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 27.1

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

   8. Simplified 27.1%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 27.1

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

   11. Simplified 27.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 48.0

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

   8. Simplified 48.0%

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

4. Final simplification 35.0%

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

Alternative 13: 35.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-5)
   (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)) <= 5e-5) {
		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)) <= 5e-5)
		tmp = fma(im, Float64(Float64(im * 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], 5e-5], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $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)) < 5.00000000000000024e-5

   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.2

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

   8. Simplified 37.2%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 37.3

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

   11. Simplified 37.3%

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

   if 5.00000000000000024e-5 < (*.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 37.9

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

   5. Simplified 37.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 28.2

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

   8. Simplified 28.2%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 28.1

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

   11. Simplified 28.1%

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

4. Final simplification 34.8%

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

Alternative 14: 35.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.00000000000000024e-5

   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.2

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

   8. Simplified 37.2%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 37.3

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

   11. Simplified 37.3%

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

   if 5.00000000000000024e-5 < (*.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 37.9

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

   5. Simplified 37.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 28.2

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

   8. Simplified 28.2%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 28.2

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

   10. Applied egg-rr 28.2%

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

   11. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 28.3

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

   13. Simplified 28.3%

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

4. Final simplification 34.9%

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

Alternative 15: 34.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 42.1

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

   5. Simplified 42.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 27.1

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

   8. Simplified 27.1%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 27.1

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

   11. Simplified 27.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 43.2

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

   8. Simplified 43.2%

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

4. Final simplification 33.1%

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

Alternative 16: 33.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.00000000000000024e-5

   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.2

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

   8. Simplified 37.2%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 37.3

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

   11. Simplified 37.3%

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

   if 5.00000000000000024e-5 < (*.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 37.9

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

   5. Simplified 37.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 21.4

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

   8. Simplified 21.4%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 21.4

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

   11. Simplified 21.4%

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

4. Final simplification 33.0%

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

Alternative 17: 34.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-5)
		tmp = fma(im, Float64(Float64(im * im) * -0.16666666666666666), im);
	else
		tmp = Float64(im * fma(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], 5e-5], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.00000000000000024e-5

   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.2

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

   8. Simplified 37.2%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 37.3

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

   11. Simplified 37.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)} \]

   if 5.00000000000000024e-5 < (*.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 37.9

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Simplified 37.9%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 28.2

         \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 28.2%

      \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right)} \]

       4. associate-+r+ N/A

          \[\leadsto im \cdot \left(re \cdot \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \frac{1}{{re}^{2}}\right)}\right)\right) \]

       5. distribute-lft-in N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right)}\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto im \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{1}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + 1\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + 1\right)\right) \]

       9. +-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + 1\right)\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + 1\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + 1\right)\right) \]

       12. lft-mult-inverse N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + 1\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + 1\right)\right) \]

   11. Simplified 28.8%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot re\right) \cdot re\right)} \]

       2. +-commutative N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot re + 1\right)} \cdot re\right) \]

       3. distribute-lft1-in N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot re\right) \cdot re + re\right)} \]

       4. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(re \cdot re\right)} + re\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{2} \cdot \color{blue}{{re}^{2}} + re\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {re}^{2}, re\right)} \]

       7. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{re \cdot re}, re\right) \]

       8. lower-*.f64 22.0

          \[\leadsto im \cdot \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, re\right) \]

   14. Simplified 22.0%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(0.5, re \cdot re, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(im, \left(im \cdot im\right) \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5, re \cdot re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 1.0) (fma im re im) (* re (* 0.5 (* re im)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 1.0) {
		tmp = fma(im, re, im);
	} else {
		tmp = re * (0.5 * (re * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 1.0)
		tmp = fma(im, re, im);
	else
		tmp = Float64(re * Float64(0.5 * Float64(re * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 1.0], N[(im * re + im), $MachinePrecision], N[(re * N[(0.5 * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 68.6

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Simplified 68.6%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + im \cdot re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{im \cdot re + im} \]

      2. lower-fma.f64 31.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

   8. Simplified 31.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot e^{re}} \]

      2. lower-exp.f64 75.8

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + re \cdot \left(im + \frac{1}{2} \cdot \left(im \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(im + \frac{1}{2} \cdot \left(im \cdot re\right)\right) + im} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(im \cdot re + \left(\frac{1}{2} \cdot \left(im \cdot re\right)\right) \cdot re\right)} + im \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\color{blue}{\left(im \cdot re\right) \cdot 1} + \left(\frac{1}{2} \cdot \left(im \cdot re\right)\right) \cdot re\right) + im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(im \cdot re\right) \cdot 1 + \color{blue}{\left(\left(im \cdot re\right) \cdot \frac{1}{2}\right)} \cdot re\right) + im \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(im \cdot re\right) \cdot 1 + \color{blue}{\left(im \cdot re\right) \cdot \left(\frac{1}{2} \cdot re\right)}\right) + im \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(im \cdot re\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)} + im \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} + im \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re \cdot \left(1 + \frac{1}{2} \cdot re\right), im\right)} \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, re \cdot \color{blue}{\left(\frac{1}{2} \cdot re + 1\right)}, im\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(im, \color{blue}{re \cdot \left(\frac{1}{2} \cdot re\right) + re \cdot 1}, im\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im, \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot re} + re \cdot 1, im\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(im, \color{blue}{\frac{1}{2} \cdot \left(re \cdot re\right)} + re \cdot 1, im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \frac{1}{2} \cdot \color{blue}{{re}^{2}} + re \cdot 1, im\right) \]

      14. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(im, \frac{1}{2} \cdot {re}^{2} + \color{blue}{re}, im\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {re}^{2}, re\right)}, im\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{re \cdot re}, re\right), im\right) \]

      17. lower-*.f64 41.1

          \[\leadsto \mathsf{fma}\left(im, \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, re\right), im\right) \]

   8. Simplified 41.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(0.5, re \cdot re, re\right), im\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot re\right) \cdot re\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot re\right)\right) \cdot re} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(im \cdot re\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(im \cdot re\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot re\right)\right)} \]

       7. lower-*.f64 27.1

          \[\leadsto re \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot re\right)}\right) \]

   11. Simplified 27.1%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \left(im \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(re \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5, re \cdot re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 2.0) (fma im re im) (* im (fma 0.5 (* re re) re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 2.0) {
		tmp = fma(im, re, im);
	} else {
		tmp = im * fma(0.5, (re * re), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 2.0)
		tmp = fma(im, re, im);
	else
		tmp = Float64(im * fma(0.5, Float64(re * re), re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(im * re + im), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 2

   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 67.7

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Simplified 67.7%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + im \cdot re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{im \cdot re + im} \]

      2. lower-fma.f64 34.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

   8. Simplified 34.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

   if 2 < (exp.f64 re)

   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 75.0

         \[\leadsto im \cdot \color{blue}{e^{re}} \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 52.7

         \[\leadsto im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 52.7%

      \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right)} \]

       4. associate-+r+ N/A

          \[\leadsto im \cdot \left(re \cdot \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \frac{1}{{re}^{2}}\right)}\right)\right) \]

       5. distribute-lft-in N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right)}\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto im \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{1}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + 1\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + 1\right)\right) \]

       9. +-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + 1\right)\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + 1\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + 1\right)\right) \]

       12. lft-mult-inverse N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + 1\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + 1\right)\right) \]

   11. Simplified 52.7%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot re\right) \cdot re\right)} \]

       2. +-commutative N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot re + 1\right)} \cdot re\right) \]

       3. distribute-lft1-in N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot re\right) \cdot re + re\right)} \]

       4. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(re \cdot re\right)} + re\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{2} \cdot \color{blue}{{re}^{2}} + re\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {re}^{2}, re\right)} \]

       7. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{re \cdot re}, re\right) \]

       8. lower-*.f64 37.8

          \[\leadsto im \cdot \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, re\right) \]

   14. Simplified 37.8%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(0.5, re \cdot re, re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.5% 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 69.5

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Simplified 69.5%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im + im \cdot re} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{im \cdot re + im} \]

   2. lower-fma.f64 27.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]

8. Simplified 27.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(im, re, im\right)} \]
9. Add Preprocessing

Alternative 21: 26.7% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

double code(double re, double im) {
	return im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im
end function

Java

public static double code(double re, double im) {
	return im;
}

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

function tmp = code(re, im)
	tmp = im;
end

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot e^{re}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot e^{re}} \]

   2. lower-exp.f64 69.5

      \[\leadsto im \cdot \color{blue}{e^{re}} \]

5. Simplified 69.5%

   \[\leadsto \color{blue}{im \cdot e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto im \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 26.6%

   \[\leadsto im \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 26.6

      \[\leadsto \color{blue}{im} \]

10. Applied egg-rr 26.6%

    \[\leadsto \color{blue}{im} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))