math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 18.5s

Alternatives: 22

Speedup: 1.0×

• 24.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999999999998:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (fma re (fma re 0.5 1.0) 1.0)))
        (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* -0.5 (* im im)))
     (if (<= t_1 -0.05)
       t_0
       (if (<= t_1 5e-26)
         (exp re)
         (if (<= t_1 0.999999999998) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (-0.5 * (im * im));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 5e-26) {
		tmp = exp(re);
	} else if (t_1 <= 0.999999999998) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(-0.5 * Float64(im * im)));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 5e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999998)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 5e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999999999998], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999999999800004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26 or 0.99999999999800004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999998:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999999999998:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ re 1.0))) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* -0.5 (* im im)))
     (if (<= t_1 -0.05)
       t_0
       (if (<= t_1 5e-26)
         (exp re)
         (if (<= t_1 0.999999999998) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (re + 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (-0.5 * (im * im));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 5e-26) {
		tmp = exp(re);
	} else if (t_1 <= 0.999999999998) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (re + 1.0);
	double t_1 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (-0.5 * (im * im));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 5e-26) {
		tmp = Math.exp(re);
	} else if (t_1 <= 0.999999999998) {
		tmp = t_0;
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (re + 1.0)
	t_1 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * (-0.5 * (im * im))
	elif t_1 <= -0.05:
		tmp = t_0
	elif t_1 <= 5e-26:
		tmp = math.exp(re)
	elif t_1 <= 0.999999999998:
		tmp = t_0
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(re + 1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(-0.5 * Float64(im * im)));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 5e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999998)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (re + 1.0);
	t_1 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * (-0.5 * (im * im));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 5e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999998)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 5e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999999999998], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999999999800004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26 or 0.99999999999800004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999998:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), t\_0, \left(t\_0 \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999999999998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (* (cos im) (+ re 1.0))))
   (if (<= t_1 (- INFINITY))
     (fma
      (fma -0.5 (* im im) 1.0)
      t_0
      (*
       (* t_0 (fma (* im im) -0.001388888888888889 0.041666666666666664))
       (* (* im im) (* im im))))
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 5e-26)
         (exp re)
         (if (<= t_1 0.999999999998) t_2 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = cos(im) * (re + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(-0.5, (im * im), 1.0), t_0, ((t_0 * fma((im * im), -0.001388888888888889, 0.041666666666666664)) * ((im * im) * (im * im))));
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 5e-26) {
		tmp = exp(re);
	} else if (t_1 <= 0.999999999998) {
		tmp = t_2;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(fma(-0.5, Float64(im * im), 1.0), t_0, Float64(Float64(t_0 * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)) * Float64(Float64(im * im) * Float64(im * im))));
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 5e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999998)
		tmp = t_2;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 5e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999999999998], t$95$2, N[Exp[re], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{720} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{24} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999999999800004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26 or 0.99999999999800004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999998:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), t\_0, \left(t\_0 \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999999999998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (fma
      (fma -0.5 (* im im) 1.0)
      t_0
      (*
       (* t_0 (fma (* im im) -0.001388888888888889 0.041666666666666664))
       (* (* im im) (* im im))))
     (if (<= t_1 -0.05)
       (cos im)
       (if (<= t_1 5e-26)
         (exp re)
         (if (<= t_1 0.999999999998) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(-0.5, (im * im), 1.0), t_0, ((t_0 * fma((im * im), -0.001388888888888889, 0.041666666666666664)) * ((im * im) * (im * im))));
	} else if (t_1 <= -0.05) {
		tmp = cos(im);
	} else if (t_1 <= 5e-26) {
		tmp = exp(re);
	} else if (t_1 <= 0.999999999998) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(fma(-0.5, Float64(im * im), 1.0), t_0, Float64(Float64(t_0 * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)) * Float64(Float64(im * im) * Float64(im * im))));
	elseif (t_1 <= -0.05)
		tmp = cos(im);
	elseif (t_1 <= 5e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999998)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$1, 5e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999999999998], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{720} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{24} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999999999800004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26 or 0.99999999999800004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \left(im \cdot im\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), t\_0, \left(t\_0 \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot t\_2\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{elif}\;t\_1 \leq 0.999999999998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (* (* im im) (* im im))))
   (if (<= t_1 (- INFINITY))
     (fma
      (fma -0.5 (* im im) 1.0)
      t_0
      (*
       (* t_0 (fma (* im im) -0.001388888888888889 0.041666666666666664))
       t_2))
     (if (<= t_1 -0.05)
       (cos im)
       (if (<= t_1 0.0)
         (* t_2 (fma re 0.041666666666666664 0.041666666666666664))
         (if (<= t_1 0.999999999998)
           (cos im)
           (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = (im * im) * (im * im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(-0.5, (im * im), 1.0), t_0, ((t_0 * fma((im * im), -0.001388888888888889, 0.041666666666666664)) * t_2));
	} else if (t_1 <= -0.05) {
		tmp = cos(im);
	} else if (t_1 <= 0.0) {
		tmp = t_2 * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else if (t_1 <= 0.999999999998) {
		tmp = cos(im);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = Float64(Float64(im * im) * Float64(im * im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(fma(-0.5, Float64(im * im), 1.0), t_0, Float64(Float64(t_0 * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)) * t_2));
	elseif (t_1 <= -0.05)
		tmp = cos(im);
	elseif (t_1 <= 0.0)
		tmp = Float64(t_2 * fma(re, 0.041666666666666664, 0.041666666666666664));
	elseif (t_1 <= 0.999999999998)
		tmp = cos(im);
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999999999998], N[Cos[im], $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{720} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{24} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999999999800004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.5

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.5%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 0.99999999999800004 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 86.3

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

   8. Applied rewrites 86.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 86.3

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

   10. Applied rewrites 86.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \left(im \cdot im\right) \cdot \left(im \cdot im\right)\\ t_2 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), t\_2, \left(t\_2 \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot t\_1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (* (* im im) (* im im)))
        (t_2 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= t_0 -0.05)
     (fma
      (fma -0.5 (* im im) 1.0)
      t_2
      (*
       (* t_2 (fma (* im im) -0.001388888888888889 0.041666666666666664))
       t_1))
     (if (<= t_0 5e-26)
       (* t_1 (fma re 0.041666666666666664 0.041666666666666664))
       (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = (im * im) * (im * im);
	double t_2 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(fma(-0.5, (im * im), 1.0), t_2, ((t_2 * fma((im * im), -0.001388888888888889, 0.041666666666666664)) * t_1));
	} else if (t_0 <= 5e-26) {
		tmp = t_1 * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(Float64(im * im) * Float64(im * im))
	t_2 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(fma(-0.5, Float64(im * im), 1.0), t_2, Float64(Float64(t_2 * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)) * t_1));
	elseif (t_0 <= 5e-26)
		tmp = Float64(t_1 * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2 + N[(N[(t$95$2 * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-26], N[(t$95$1 * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + {im}^{2} \cdot \left(\frac{-1}{720} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right) + \frac{1}{24} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.0

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, im \cdot im, 1\right), \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (* im im)
       (fma
        (* im im)
        (fma (* im im) -0.001388888888888889 0.041666666666666664)
        -0.5)
       1.0))
     (if (<= t_0 5e-26)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= 5e-26) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	elseif (t_0 <= 5e-26)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-26], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}, \frac{-1}{2}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}, \frac{-1}{2}\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}, \frac{-1}{2}\right), 1\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

      14. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]

   8. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.0

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot -0.001388888888888889\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma (* im im) (* (* im im) (* im (* im -0.001388888888888889))) 1.0))
     (if (<= t_0 5e-26)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma((im * im), ((im * im) * (im * (im * -0.001388888888888889))), 1.0);
	} else if (t_0 <= 5e-26) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(Float64(im * im), Float64(Float64(im * im) * Float64(im * Float64(im * -0.001388888888888889))), 1.0));
	elseif (t_0 <= 5e-26)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-26], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}, \frac{-1}{2}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}, \frac{-1}{2}\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}, \frac{-1}{2}\right), 1\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

      14. lower-*.f64 35.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]

   8. Applied rewrites 35.0%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{720} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{720} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2}\right)}, 1\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{720} \cdot {im}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{720} \cdot {im}^{2}\right), 1\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{720} \cdot {im}^{2}\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{-1}{720}\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{720}\right), 1\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{-1}{720}\right)\right)}, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{-1}{720}\right)\right)}, 1\right) \]

       12. lower-*.f64 35.0

           \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot -0.001388888888888889\right)}\right), 1\right) \]

   11. Applied rewrites 35.0%

       \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot -0.001388888888888889\right)\right)}, 1\right) \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.0

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot -0.001388888888888889\right)\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (* (fma re (fma re 0.5 1.0) 1.0) (fma im (* im -0.5) 1.0))
     (if (<= t_0 5e-26)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 5e-26) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 5e-26)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-26], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.8

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 33.1

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

   8. Applied rewrites 33.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.0

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (* (* -0.5 (* im im)) (fma re (fma re 0.5 1.0) 1.0))
     (if (<= t_0 5e-26)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (-0.5 * (im * im)) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (t_0 <= 5e-26) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_0 <= 5e-26)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-26], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.8

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 37.8

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 33.1

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

   11. Applied rewrites 33.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.0

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (* (+ re 1.0) (fma -0.5 (* im im) 1.0))
     (if (<= t_0 5e-26)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (re + 1.0) * fma(-0.5, (im * im), 1.0);
	} else if (t_0 <= 5e-26) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(re + 1.0) * fma(-0.5, Float64(im * im), 1.0));
	elseif (t_0 <= 5e-26)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(re + 1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-26], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 28.4

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

   8. Applied rewrites 28.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 37.0

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* im (* im -0.5))
     (if (<= t_0 2.0)
       (fma re (fma re 0.5 1.0) 1.0)
       (* (fma re 0.16666666666666666 0.5) (* re re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = im * (im * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = fma(re, 0.16666666666666666, 0.5) * (re * re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (t_0 <= 2.0)
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	else
		tmp = Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.9

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

   11. Applied rewrites 28.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.5

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

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 67.5

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

   11. Applied rewrites 67.5%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* im (* im -0.5))
     (if (<= t_0 2.0)
       (fma re (fma re 0.5 1.0) 1.0)
       (* 0.16666666666666666 (* re (* re re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = im * (im * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = 0.16666666666666666 * (re * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (t_0 <= 2.0)
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	else
		tmp = Float64(0.16666666666666666 * Float64(re * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.9

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

   11. Applied rewrites 28.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.5

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

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 67.5

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

   11. Applied rewrites 67.5%

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

Alternative 15: 50.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 5e-26)
   (* im (* im -0.5))
   (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 5e-26) {
		tmp = im * (im * -0.5);
	} else {
		tmp = re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 5e-26)
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 5e-26], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 5e-26)
   (* im (* im -0.5))
   (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) * cos(im)) <= 5e-26) {
		tmp = im * (im * -0.5);
	} else {
		tmp = 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) * cos(im)) <= 5e-26)
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = 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[Cos[im], $MachinePrecision]), $MachinePrecision], 5e-26], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

Alternative 17: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 5e-26)
   (* im (* im -0.5))
   (fma re (* re (* re 0.16666666666666666)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 5e-26) {
		tmp = im * (im * -0.5);
	} else {
		tmp = fma(re, (re * (re * 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 5e-26)
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = fma(re, Float64(re * Float64(re * 0.16666666666666666)), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 5e-26], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 73.1

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

   11. Applied rewrites 73.1%

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

Alternative 18: 47.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.9

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

   11. Applied rewrites 28.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.9

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

   8. Applied rewrites 69.9%

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

Alternative 19: 39.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 5e-26) (* im (* im -0.5)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 5e-26) {
		tmp = im * (im * -0.5);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 5d-26) then
        tmp = im * (im * (-0.5d0))
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 5e-26) {
		tmp = im * (im * -0.5);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 5e-26:
		tmp = im * (im * -0.5)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 5e-26)
		tmp = Float64(im * Float64(im * -0.5));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 5e-26)
		tmp = im * (im * -0.5);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 5e-26], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 5.00000000000000019e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \frac{-1}{2}} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if 5.00000000000000019e-26 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.3

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. lower-+.f64 52.8

         \[\leadsto \color{blue}{1 + re} \]

   8. Applied rewrites 52.8%

      \[\leadsto \color{blue}{1 + re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00028:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+95}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))
   (if (<= re -0.0022)
     (exp re)
     (if (<= re 0.00028) t_0 (if (<= re 1.45e+95) (exp re) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (re <= -0.0022) {
		tmp = exp(re);
	} else if (re <= 0.00028) {
		tmp = t_0;
	} else if (re <= 1.45e+95) {
		tmp = exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	tmp = 0.0
	if (re <= -0.0022)
		tmp = exp(re);
	elseif (re <= 0.00028)
		tmp = t_0;
	elseif (re <= 1.45e+95)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0022], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.00028], t$95$0, If[LessEqual[re, 1.45e+95], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00220000000000000013 or 2.7999999999999998e-4 < re < 1.45000000000000007e95

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 96.3

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00220000000000000013 < re < 2.7999999999999998e-4 or 1.45000000000000007e95 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \cos im \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \cos im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \cos im \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \cos im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \cos im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \cos im \]

      7. lower-fma.f64 99.5

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00028:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+95}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.0% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. lower-exp.f64 75.5

      \[\leadsto \color{blue}{e^{re}} \]

5. Applied rewrites 75.5%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. lower-+.f64 33.1

      \[\leadsto \color{blue}{1 + re} \]

8. Applied rewrites 33.1%

   \[\leadsto \color{blue}{1 + re} \]

9. Final simplification 33.1%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 22: 28.6% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. lower-exp.f64 75.5

      \[\leadsto \color{blue}{e^{re}} \]

5. Applied rewrites 75.5%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 32.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))