math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.4s

Alternatives: 28

Speedup: 1.0×

• 23.9% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (fma im (* im -0.5) 1.0))
     (if (<= t_0 -0.01)
       (* (cos im) (fma re (fma re 0.5 1.0) 1.0))
       (if (<= t_0 2e-26)
         (exp re)
         (if (<= t_0 0.9999999)
           (* (cos im) (fma (* re re) 0.5 (+ re 1.0)))
           (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= -0.01) {
		tmp = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (t_0 <= 2e-26) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999) {
		tmp = cos(im) * fma((re * re), 0.5, (re + 1.0));
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= -0.01)
		tmp = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_0 <= 2e-26)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999)
		tmp = Float64(cos(im) * fma(Float64(re * re), 0.5, Float64(re + 1.0)));
	else
		tmp = exp(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, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.5 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $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 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. *-lowering-*.f64 N/A

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

      5. exp-lowering-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. accelerator-lowering-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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 98.0

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

   5. Simplified 98.0%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.999999900000000053 < (*.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. exp-lowering-exp.f64 99.7

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

   5. Simplified 99.7%

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

   if 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999900000000053

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 96.3

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

   5. Simplified 96.3%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 96.3

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

   7. Applied egg-rr 96.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

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) (fma im (* im -0.5) 1.0))
     (if (<= t_1 -0.01)
       t_0
       (if (<= t_1 2e-26) (exp re) (if (<= t_1 0.9999999) 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) * fma(im, (im * -0.5), 1.0);
	} else if (t_1 <= -0.01) {
		tmp = t_0;
	} else if (t_1 <= 2e-26) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999) {
		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) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_1 <= -0.01)
		tmp = t_0;
	elseif (t_1 <= 2e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999)
		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[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.01], t$95$0, If[LessEqual[t$95$1, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999], 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. *-lowering-*.f64 N/A

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

      5. exp-lowering-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. accelerator-lowering-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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999900000000053

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 97.3

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

   5. Simplified 97.3%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.999999900000000053 < (*.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. exp-lowering-exp.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

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) (fma im (* im -0.5) 1.0))
     (if (<= t_1 -0.01)
       t_0
       (if (<= t_1 2e-26) (exp re) (if (<= t_1 0.999) 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) * fma(im, (im * -0.5), 1.0);
	} else if (t_1 <= -0.01) {
		tmp = t_0;
	} else if (t_1 <= 2e-26) {
		tmp = exp(re);
	} else if (t_1 <= 0.999) {
		tmp = t_0;
	} else {
		tmp = 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) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_1 <= -0.01)
		tmp = t_0;
	elseif (t_1 <= 2e-26)
		tmp = exp(re);
	elseif (t_1 <= 0.999)
		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 + 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[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.01], t$95$0, If[LessEqual[t$95$1, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999], 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. *-lowering-*.f64 N/A

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

      5. exp-lowering-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. accelerator-lowering-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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   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. +-lowering-+.f64 97.0

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

   5. Simplified 97.0%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.998999999999999999 < (*.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. exp-lowering-exp.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right), re \cdot -0.5\right), re\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (cos im) (+ re 1.0))))
   (if (<= t_0 (- INFINITY))
     (fma
      (* im im)
      (fma
       re
       (* im (* im (fma (* im im) -0.001388888888888889 0.041666666666666664)))
       (* re -0.5))
      re)
     (if (<= t_0 -0.01)
       t_1
       (if (<= t_0 2e-26) (exp re) (if (<= t_0 0.999) t_1 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * (re + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((im * im), fma(re, (im * (im * fma((im * im), -0.001388888888888889, 0.041666666666666664))), (re * -0.5)), re);
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 2e-26) {
		tmp = exp(re);
	} else if (t_0 <= 0.999) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(im * im), fma(re, Float64(im * Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664))), Float64(re * -0.5)), re);
	elseif (t_0 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 2e-26)
		tmp = exp(re);
	elseif (t_0 <= 0.999)
		tmp = t_1;
	else
		tmp = exp(re);
	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[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * -0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], If[LessEqual[t$95$0, -0.01], t$95$1, If[LessEqual[t$95$0, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.999], t$95$1, 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

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 5.3

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 5.3

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

   8. Simplified 5.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 94.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   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. +-lowering-+.f64 97.0

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

   5. Simplified 97.0%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.998999999999999999 < (*.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. exp-lowering-exp.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right), re \cdot -0.5\right), re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right), re \cdot -0.5\right), re\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (* im im)
      (fma
       re
       (* im (* im (fma (* im im) -0.001388888888888889 0.041666666666666664)))
       (* re -0.5))
      re)
     (if (<= t_0 -0.01)
       (cos im)
       (if (<= t_0 2e-26) (exp re) (if (<= t_0 0.999) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((im * im), fma(re, (im * (im * fma((im * im), -0.001388888888888889, 0.041666666666666664))), (re * -0.5)), re);
	} else if (t_0 <= -0.01) {
		tmp = cos(im);
	} else if (t_0 <= 2e-26) {
		tmp = exp(re);
	} else if (t_0 <= 0.999) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(im * im), fma(re, Float64(im * Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664))), Float64(re * -0.5)), re);
	elseif (t_0 <= -0.01)
		tmp = cos(im);
	elseif (t_0 <= 2e-26)
		tmp = exp(re);
	elseif (t_0 <= 0.999)
		tmp = cos(im);
	else
		tmp = exp(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, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * -0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.999], 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

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 5.3

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 5.3

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

   8. Simplified 5.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 94.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 95.8

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

   5. Simplified 95.8%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.998999999999999999 < (*.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. exp-lowering-exp.f64 99.4

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

   5. Simplified 99.4%

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

Alternative 7: 78.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right)\right), re \cdot -0.5\right), re\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (* im im)
      (fma
       re
       (* im (* im (fma (* im im) -0.001388888888888889 0.041666666666666664)))
       (* re -0.5))
      re)
     (if (<= t_0 -0.01)
       (cos im)
       (if (<= t_0 0.0)
         (* (+ re 1.0) (* 0.041666666666666664 (* (* im im) (* im im))))
         (if (<= t_0 0.999)
           (cos im)
           (*
            (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
            (fma
             im
             (* im (fma im (* im 0.041666666666666664) -0.5))
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((im * im), fma(re, (im * (im * fma((im * im), -0.001388888888888889, 0.041666666666666664))), (re * -0.5)), re);
	} else if (t_0 <= -0.01) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (re + 1.0) * (0.041666666666666664 * ((im * im) * (im * im)));
	} else if (t_0 <= 0.999) {
		tmp = cos(im);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, (im * fma(im, (im * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(im * im), fma(re, Float64(im * Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664))), Float64(re * -0.5)), re);
	elseif (t_0 <= -0.01)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(re + 1.0) * Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im))));
	elseif (t_0 <= 0.999)
		tmp = cos(im);
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), -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, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * -0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(re + 1.0), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999], N[Cos[im], $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + -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

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 5.3

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 5.3

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

   8. Simplified 5.3%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 94.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 93.6

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

   5. Simplified 93.6%

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

   if -0.0100000000000000002 < (*.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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 35.7

          \[\leadsto \left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 35.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 87.8

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

   5. Simplified 87.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot im\right) \cdot im} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{im \cdot \left(\frac{1}{24} \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{-1}{2}\right)}, 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      15. *-lowering-*.f64 91.5

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

   8. Simplified 91.5%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (* (cos im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= t_0 2e-26)
       (exp re)
       (if (<= t_0 0.999)
         (*
          (cos im)
          (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0)))
         (*
          (exp re)
          (fma im (* im (fma (* im im) 0.041666666666666664 -0.5)) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 2e-26) {
		tmp = exp(re);
	} else if (t_0 <= 0.999) {
		tmp = cos(im) * (re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	} else {
		tmp = exp(re) * fma(im, (im * fma((im * im), 0.041666666666666664, -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 2e-26)
		tmp = exp(re);
	elseif (t_0 <= 0.999)
		tmp = Float64(cos(im) * Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0)));
	else
		tmp = Float64(exp(re) * fma(im, Float64(im * fma(Float64(im * im), 0.041666666666666664, -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, -0.01], 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[t$95$0, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.999], N[(N[Cos[im], $MachinePrecision] * N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 99.3

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

   5. Simplified 99.3%

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

   if 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 97.8

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

   5. Simplified 97.8%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-fma.f64 97.8

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

   7. Applied egg-rr 97.8%

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

   if 0.998999999999999999 < (*.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. distribute-rgt-out N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re} \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (* (cos im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= t_0 2e-26)
       (exp re)
       (if (<= t_0 0.9999999)
         (*
          (cos im)
          (+ re (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0)))
         (exp re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 2e-26) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999) {
		tmp = cos(im) * (re + fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0));
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 2e-26)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999)
		tmp = Float64(cos(im) * Float64(re + fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0)));
	else
		tmp = exp(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, -0.01], 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[t$95$0, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999], N[(N[Cos[im], $MachinePrecision] * N[(re + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.999999900000000053 < (*.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. exp-lowering-exp.f64 99.7

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

   5. Simplified 99.7%

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

   if 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999900000000053

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 97.6

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

   5. Simplified 97.6%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-fma.f64 97.6

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

   7. Applied egg-rr 97.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1
         (*
          (cos im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))
   (if (<= t_0 -0.01)
     t_1
     (if (<= t_0 2e-26) (exp re) (if (<= t_0 0.9999999) t_1 (exp re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 2e-26) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 2e-26)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999)
		tmp = t_1;
	else
		tmp = exp(re);
	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[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], t$95$1, If[LessEqual[t$95$0, 2e-26], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999], t$95$1, N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-26 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999900000000053

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 94.3

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

   5. Simplified 94.3%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26 or 0.999999900000000053 < (*.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. exp-lowering-exp.f64 99.7

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;\mathsf{fma}\left(re, \frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.08333333333333333, -0.5\right), 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (fma
      (* im im)
      (fma
       (* im im)
       (fma (* im im) -0.001388888888888889 0.041666666666666664)
       -0.5)
      1.0)
     (if (<= t_0 0.0)
       (* (+ re 1.0) (* 0.041666666666666664 (* (* im im) (* im im))))
       (if (<= t_0 0.9995)
         (fma re (/ 1.0 (fma re (fma re 0.08333333333333333 -0.5) 1.0)) 1.0)
         (*
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
          (fma im (* im (fma im (* im 0.041666666666666664) -0.5)) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (re + 1.0) * (0.041666666666666664 * ((im * im) * (im * im)));
	} else if (t_0 <= 0.9995) {
		tmp = fma(re, (1.0 / fma(re, fma(re, 0.08333333333333333, -0.5), 1.0)), 1.0);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, (im * fma(im, (im * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(re + 1.0) * Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im))));
	elseif (t_0 <= 0.9995)
		tmp = fma(re, Float64(1.0 / fma(re, fma(re, 0.08333333333333333, -0.5), 1.0)), 1.0);
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), -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, -0.01], 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], If[LessEqual[t$95$0, 0.0], N[(N[(re + 1.0), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(re * N[(1.0 / N[(re * N[(re * 0.08333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 28.7

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

   8. Simplified 28.7%

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

   if -0.0100000000000000002 < (*.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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 35.7

          \[\leadsto \left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 35.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 21.7

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

   8. Simplified 21.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 21.7

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

   10. Applied egg-rr 21.7%

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

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{\color{blue}{re \cdot \left(\frac{1}{12} \cdot re - \frac{1}{2}\right) + 1}}, 1\right) \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{\color{blue}{\mathsf{fma}\left(re, \frac{1}{12} \cdot re - \frac{1}{2}, 1\right)}}, 1\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{\mathsf{fma}\left(re, \color{blue}{\frac{1}{12} \cdot re + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{12}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}, 1\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{\mathsf{fma}\left(re, re \cdot \frac{1}{12} + \color{blue}{\frac{-1}{2}}, 1\right)}, 1\right) \]

       6. accelerator-lowering-fma.f64 22.4

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{\mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.08333333333333333, -0.5\right)}, 1\right)}, 1\right) \]

   13. Simplified 22.4%

       \[\leadsto \mathsf{fma}\left(re, \frac{1}{\color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.08333333333333333, -0.5\right), 1\right)}}, 1\right) \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 87.8

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

   5. Simplified 87.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot im\right) \cdot im} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{im \cdot \left(\frac{1}{24} \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left(im, \frac{1}{24} \cdot im, \frac{-1}{2}\right)}, 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      15. *-lowering-*.f64 91.6

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

   8. Simplified 91.6%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (fma
      (* im im)
      (fma
       (* im im)
       (fma (* im im) -0.001388888888888889 0.041666666666666664)
       -0.5)
      1.0)
     (if (<= t_0 2e-26)
       (* (+ re 1.0) (* 0.041666666666666664 (* (* im im) (* im im))))
       (fma
        (* (* re re) (fma (* re re) 0.027777777777777776 -0.25))
        (/ 1.0 (fma re 0.16666666666666666 -0.5))
        (+ re 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= 2e-26) {
		tmp = (re + 1.0) * (0.041666666666666664 * ((im * im) * (im * im)));
	} else {
		tmp = fma(((re * re) * fma((re * re), 0.027777777777777776, -0.25)), (1.0 / fma(re, 0.16666666666666666, -0.5)), (re + 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	elseif (t_0 <= 2e-26)
		tmp = Float64(Float64(re + 1.0) * Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = fma(Float64(Float64(re * re) * fma(Float64(re * re), 0.027777777777777776, -0.25)), Float64(1.0 / fma(re, 0.16666666666666666, -0.5)), Float64(re + 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, -0.01], 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], If[LessEqual[t$95$0, 2e-26], N[(N[(re + 1.0), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 28.7

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

   8. Simplified 28.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 34.6

          \[\leadsto \left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 34.6%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. flip-+ N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 73.1%

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

Alternative 13: 55.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (fma
      (* im im)
      (fma
       (* im im)
       (fma (* im im) -0.001388888888888889 0.041666666666666664)
       -0.5)
      1.0)
     (if (<= t_0 2e-26)
       (* (+ re 1.0) (* 0.041666666666666664 (* (* im im) (* im im))))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= 2e-26) {
		tmp = (re + 1.0) * (0.041666666666666664 * ((im * im) * (im * im)));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	elseif (t_0 <= 2e-26)
		tmp = Float64(Float64(re + 1.0) * Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 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, -0.01], 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], If[LessEqual[t$95$0, 2e-26], N[(N[(re + 1.0), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 28.7

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

   8. Simplified 28.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 34.6

          \[\leadsto \left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 34.6%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

Alternative 14: 55.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (* (fma im (* im -0.5) 1.0) (fma re (fma re 0.5 1.0) 1.0))
     (if (<= t_0 2e-26)
       (* (+ re 1.0) (* 0.041666666666666664 (* (* im im) (* im im))))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma(im, (im * -0.5), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (t_0 <= 2e-26) {
		tmp = (re + 1.0) * (0.041666666666666664 * ((im * im) * (im * im)));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_0 <= 2e-26)
		tmp = Float64(Float64(re + 1.0) * Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 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, -0.01], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-26], N[(N[(re + 1.0), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   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. *-lowering-*.f64 N/A

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

      5. exp-lowering-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. accelerator-lowering-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. *-lowering-*.f64 30.7

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

   5. Simplified 30.7%

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

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 27.7

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

   8. Simplified 27.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 34.6

          \[\leadsto \left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 34.6%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (* im (* im (fma re -0.5 -0.5)))
     (if (<= t_0 2e-26)
       (* (+ re 1.0) (* 0.041666666666666664 (* (* im im) (* im im))))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = im * (im * fma(re, -0.5, -0.5));
	} else if (t_0 <= 2e-26) {
		tmp = (re + 1.0) * (0.041666666666666664 * ((im * im) * (im * im)));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(im * Float64(im * fma(re, -0.5, -0.5)));
	elseif (t_0 <= 2e-26)
		tmp = Float64(Float64(re + 1.0) * Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im))));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 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, -0.01], N[(im * N[(im * N[(re * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-26], N[(N[(re + 1.0), $MachinePrecision] * N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   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. +-lowering-+.f64 71.9

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

   5. Simplified 71.9%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 0.5

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

   8. Simplified 0.5%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 23.3

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

   11. Simplified 23.3%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 23.3

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

   14. Simplified 23.3%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 34.6

          \[\leadsto \left(re + 1\right) \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 34.6%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

Alternative 16: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.01)
     (* im (* im (fma re -0.5 -0.5)))
     (if (<= t_0 2e-26)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = im * (im * fma(re, -0.5, -0.5));
	} else if (t_0 <= 2e-26) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(im * Float64(im * fma(re, -0.5, -0.5)));
	elseif (t_0 <= 2e-26)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 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, -0.01], N[(im * N[(im * N[(re * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-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 * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   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. +-lowering-+.f64 71.9

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

   5. Simplified 71.9%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 0.5

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

   8. Simplified 0.5%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 23.3

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

   11. Simplified 23.3%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 23.3

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

   14. Simplified 23.3%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 2.2

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

   5. Simplified 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.8

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

   8. Simplified 1.8%

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

   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. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-fma.f64 33.0

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

   11. Simplified 33.0%

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

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

Alternative 17: 49.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 5e-127)
     (* im (* im (fma re -0.5 -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 <= 5e-127) {
		tmp = im * (im * fma(re, -0.5, -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 <= 5e-127)
		tmp = Float64(im * Float64(im * fma(re, -0.5, -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, 5e-127], N[(im * N[(im * N[(re * -0.5 + -0.5), $MachinePrecision]), $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)) < 4.9999999999999997e-127

   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. +-lowering-+.f64 37.6

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

   5. Simplified 37.6%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.2

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

   8. Simplified 1.2%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 12.7

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

   11. Simplified 12.7%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 17.0

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

   14. Simplified 17.0%

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

   if 4.9999999999999997e-127 < (*.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. exp-lowering-exp.f64 71.4

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

   5. Simplified 71.4%

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

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 70.0

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

   8. Simplified 70.0%

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

   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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

   5. Simplified 71.3%

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

   6. Taylor expanded in im 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

   8. Simplified 71.3%

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

   9. 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. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 71.3

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

   11. Simplified 71.3%

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

Alternative 18: 49.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 5e-127)
     (* im (* im (fma re -0.5 -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 <= 5e-127) {
		tmp = im * (im * fma(re, -0.5, -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 <= 5e-127)
		tmp = Float64(im * Float64(im * fma(re, -0.5, -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, 5e-127], N[(im * N[(im * N[(re * -0.5 + -0.5), $MachinePrecision]), $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)) < 4.9999999999999997e-127

   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. +-lowering-+.f64 37.6

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

   5. Simplified 37.6%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.2

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

   8. Simplified 1.2%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 12.7

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

   11. Simplified 12.7%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 17.0

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

   14. Simplified 17.0%

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

   if 4.9999999999999997e-127 < (*.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. exp-lowering-exp.f64 71.4

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

   5. Simplified 71.4%

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

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 70.0

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

   8. Simplified 70.0%

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

   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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

   5. Simplified 71.3%

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

   6. Taylor expanded in im 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

   8. Simplified 71.3%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 71.3

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

   11. Simplified 71.3%

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

Alternative 19: 45.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 2e-26)
     (fma -0.5 (* im im) 1.0)
     (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 <= 2e-26) {
		tmp = fma(-0.5, (im * im), 1.0);
	} 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 <= 2e-26)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	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, 2e-26], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $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)) < 2.0000000000000001e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 36.9

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

   5. Simplified 36.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 9.7

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

   8. Simplified 9.7%

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

   if 2.0000000000000001e-26 < (*.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. exp-lowering-exp.f64 71.1

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

   5. Simplified 71.1%

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

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 70.7

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

   8. Simplified 70.7%

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

   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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

   5. Simplified 71.3%

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

   6. Taylor expanded in im 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

   8. Simplified 71.3%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 71.3

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

   11. Simplified 71.3%

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

Alternative 20: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2e-26)
   (* (+ re 1.0) (* (* 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)) <= 2e-26) {
		tmp = (re + 1.0) * ((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)) <= 2e-26)
		tmp = Float64(Float64(re + 1.0) * Float64(Float64(im * 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], 2e-26], N[(N[(re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -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)) < 2.0000000000000001e-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. +-lowering-+.f64 37.3

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

   5. Simplified 37.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.2

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

   8. Simplified 1.2%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 12.6

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

   11. Simplified 12.6%

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

   12. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 22.5

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

   14. Simplified 22.5%

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

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, re \cdot 0.16666666666666666, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2e-26)
   (* (+ re 1.0) (* (* im im) -0.5))
   (fma re (fma re (* re 0.16666666666666666) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2e-26) {
		tmp = (re + 1.0) * ((im * im) * -0.5);
	} else {
		tmp = fma(re, fma(re, (re * 0.16666666666666666), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 2e-26)
		tmp = Float64(Float64(re + 1.0) * Float64(Float64(im * im) * -0.5));
	else
		tmp = fma(re, fma(re, Float64(re * 0.16666666666666666), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 2e-26], N[(N[(re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 37.3

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

   5. Simplified 37.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.2

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

   8. Simplified 1.2%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 12.6

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

   11. Simplified 12.6%

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

   12. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 22.5

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

   14. Simplified 22.5%

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

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 70.3

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

   11. Simplified 70.3%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, re \cdot 0.16666666666666666, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 49.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, re \cdot 0.16666666666666666, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2e-26)
   (* im (* im (fma re -0.5 -0.5)))
   (fma re (fma re (* re 0.16666666666666666) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2e-26) {
		tmp = im * (im * fma(re, -0.5, -0.5));
	} else {
		tmp = fma(re, fma(re, (re * 0.16666666666666666), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 2e-26)
		tmp = Float64(im * Float64(im * fma(re, -0.5, -0.5)));
	else
		tmp = fma(re, fma(re, Float64(re * 0.16666666666666666), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 2e-26], N[(im * N[(im * N[(re * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-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. +-lowering-+.f64 37.3

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

   5. Simplified 37.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.2

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

   8. Simplified 1.2%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 12.6

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

   11. Simplified 12.6%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 16.9

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

   14. Simplified 16.9%

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

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

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

   8. Simplified 71.1%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 70.3

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

   11. Simplified 70.3%

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

Alternative 23: 49.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2e-26)
   (* im (* im (fma re -0.5 -0.5)))
   (fma re (* re (* re 0.16666666666666666)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2e-26) {
		tmp = im * (im * fma(re, -0.5, -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)) <= 2e-26)
		tmp = Float64(im * Float64(im * fma(re, -0.5, -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], 2e-26], N[(im * N[(im * N[(re * -0.5 + -0.5), $MachinePrecision]), $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)) < 2.0000000000000001e-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. +-lowering-+.f64 37.3

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

   5. Simplified 37.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 1.2

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

   8. Simplified 1.2%

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

   9. Taylor expanded in im around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 12.6

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

   11. Simplified 12.6%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(1 + re\right)\right) \cdot \frac{-1}{2}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(1 + re\right) \cdot \frac{-1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto im \cdot \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(re + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(re \cdot \frac{-1}{2} + 1 \cdot \frac{-1}{2}\right)}\right) \]

       10. metadata-eval N/A

           \[\leadsto im \cdot \left(im \cdot \left(re \cdot \frac{-1}{2} + \color{blue}{\frac{-1}{2}}\right)\right) \]

       11. accelerator-lowering-fma.f64 16.9

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(re, -0.5, -0.5\right)}\right) \]

   14. Simplified 16.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \mathsf{fma}\left(re, -0.5, -0.5\right)\right)} \]

   if 2.0000000000000001e-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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 90.2

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.1

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   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. *-lowering-*.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. *-lowering-*.f64 69.3

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}, 1\right) \]

   11. Simplified 69.3%

       \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(re \cdot 0.16666666666666666\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 42.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2e-26)
   (fma -0.5 (* im im) 1.0)
   (fma re (fma re 0.5 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2e-26) {
		tmp = fma(-0.5, (im * im), 1.0);
	} 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)) <= 2e-26)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	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], 2e-26], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $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)) < 2.0000000000000001e-26

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 36.9

         \[\leadsto \color{blue}{\cos im} \]

   5. Simplified 36.9%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. *-lowering-*.f64 9.7

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Simplified 9.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 2.0000000000000001e-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. exp-lowering-exp.f64 80.3

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{e^{re}} \]

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 67.4

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \]

   8. Simplified 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 33.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2e-26) (fma -0.5 (* im im) 1.0) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2e-26) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 2e-26)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 2e-26], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 2.0000000000000001e-26

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 36.9

         \[\leadsto \color{blue}{\cos im} \]

   5. Simplified 36.9%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. *-lowering-*.f64 9.7

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Simplified 9.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 2.0000000000000001e-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. exp-lowering-exp.f64 80.3

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re + 1} \]

      2. +-lowering-+.f64 49.3

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 49.3%

      \[\leadsto \color{blue}{re + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00096:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 11:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00096)
   (exp re)
   (if (<= re 11.0)
     (* (cos im) (fma (* re re) 0.5 (+ re 1.0)))
     (if (<= re 1e+103)
       (exp re)
       (* (cos im) (* re (* 0.16666666666666666 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00096) {
		tmp = exp(re);
	} else if (re <= 11.0) {
		tmp = cos(im) * fma((re * re), 0.5, (re + 1.0));
	} else if (re <= 1e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00096)
		tmp = exp(re);
	elseif (re <= 11.0)
		tmp = Float64(cos(im) * fma(Float64(re * re), 0.5, Float64(re + 1.0)));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00096], N[Exp[re], $MachinePrecision], If[LessEqual[re, 11.0], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.5 + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.60000000000000024e-4 or 11 < re < 1e103

   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. exp-lowering-exp.f64 93.4

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if -9.60000000000000024e-4 < re < 11

   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. accelerator-lowering-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. accelerator-lowering-fma.f64 99.0

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \cdot \cos im \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \frac{1}{2}\right) + re \cdot 1\right)} + 1\right) \cdot \cos im \]

      2. *-rgt-identity N/A

         \[\leadsto \left(\left(re \cdot \left(re \cdot \frac{1}{2}\right) + \color{blue}{re}\right) + 1\right) \cdot \cos im \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \frac{1}{2}\right) + \left(re + 1\right)\right)} \cdot \cos im \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right) \cdot \frac{1}{2}} + \left(re + 1\right)\right) \cdot \cos im \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \frac{1}{2}, re + 1\right)} \cdot \cos im \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{2}, re + 1\right) \cdot \cos im \]

      7. +-lowering-+.f64 99.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, 0.5, \color{blue}{re + 1}\right) \cdot \cos im \]

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)} \cdot \cos im \]

   if 1e103 < 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]

   6. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot \cos im \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot re\right)}\right) \cdot \cos im \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{{re}^{2}} \cdot re\right)\right) \cdot \cos im \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot re\right)} \cdot \cos im \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot \cos im \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot \cos im \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right)}\right) \cdot \cos im \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \cos im \]

      8. *-lowering-*.f64 100.0

         \[\leadsto \left(re \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \cos im \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot \cos im \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00096:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 11:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re \cdot re, 0.5, re + 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 30.0% 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. exp-lowering-exp.f64 65.9

      \[\leadsto \color{blue}{e^{re}} \]

5. Simplified 65.9%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{re + 1} \]

   2. +-lowering-+.f64 26.8

      \[\leadsto \color{blue}{re + 1} \]

8. Simplified 26.8%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 28: 29.5% 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 re around 0

   \[\leadsto \color{blue}{\cos im} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 52.4

      \[\leadsto \color{blue}{\cos im} \]

5. Simplified 52.4%

   \[\leadsto \color{blue}{\cos im} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 26.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))