math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 20.1s

Alternatives: 20

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999996580522:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

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 -0.02)
       t_0
       (if (<= t_1 0.0)
         (exp re)
         (if (<= t_1 0.9999999996580522) 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 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999996580522) {
		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 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999996580522)
		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, -0.02], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999996580522], 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)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.9999999996580522

   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 99.1

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

   5. Simplified 99.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.9999999996580522 < (*.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 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999996580522:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999996580522:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

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 -0.02)
       t_0
       (if (<= t_1 0.0)
         (exp re)
         (if (<= t_1 0.9999999996580522) 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 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999996580522) {
		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 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999996580522)
		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, -0.02], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999996580522], 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)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.9999999996580522

   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 98.3

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

   5. Simplified 98.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.9999999996580522 < (*.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 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999996580522:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999996580522:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (cos im) (+ re 1.0))))
   (if (<= t_0 (- INFINITY))
     (*
      (* re (* re re))
      (fma (* im im) -0.08333333333333333 0.16666666666666666))
     (if (<= t_0 -0.02)
       t_1
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 0.9999999996580522) 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 = (re * (re * re)) * fma((im * im), -0.08333333333333333, 0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999996580522) {
		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 = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999996580522)
		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[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999996580522], 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 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%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.8

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

   8. Simplified 93.8%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. *-commutative N/A

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

       11. +-commutative N/A

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

       12. distribute-lft1-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + \frac{1}{6}\right) \]

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

       20. metadata-eval 93.8

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right) \]

   11. Simplified 93.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.9999999996580522

   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 98.3

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

   5. Simplified 98.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.9999999996580522 < (*.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 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999996580522:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999996580522:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* re (* re re))
      (fma (* im im) -0.08333333333333333 0.16666666666666666))
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 0.9999999996580522) (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 = (re * (re * re)) * fma((im * im), -0.08333333333333333, 0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999996580522) {
		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 = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999996580522)
		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[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999996580522], 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 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%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.8

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

   8. Simplified 93.8%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. *-commutative N/A

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

       11. +-commutative N/A

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

       12. distribute-lft1-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + \frac{1}{6}\right) \]

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

       20. metadata-eval 93.8

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right) \]

   11. Simplified 93.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.9999999996580522

   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 97.2

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

   5. Simplified 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.9999999996580522 < (*.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 99.5

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

   5. Simplified 99.5%

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

Alternative 6: 56.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= t_0 -0.02)
     (* (fma im (* im -0.5) 1.0) t_1)
     (if (<= t_0 0.0) (* 0.041666666666666664 (* (* im im) (* im im))) t_1))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(im, (im * -0.5), 1.0) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = t_1;
	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[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

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

   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 31.1

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

   5. Simplified 31.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 29.4

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

   8. Simplified 29.4%

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

   if -0.0200000000000000004 < (*.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

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

      1. cos-lowering-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 42.9

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

   11. Simplified 42.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in 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 75.5

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

   8. Simplified 75.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (*
      (fma im (* im -0.5) 1.0)
      (* (fma re 0.16666666666666666 0.5) (* re re)))
     (if (<= t_0 0.0)
       (* 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 <= -0.05) {
		tmp = fma(im, (im * -0.5), 1.0) * (fma(re, 0.16666666666666666, 0.5) * (re * re));
	} else if (t_0 <= 0.0) {
		tmp = 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 <= -0.05)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)));
	elseif (t_0 <= 0.0)
		tmp = 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, -0.05], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $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)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-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 31.7

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

   5. Simplified 31.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 29.9

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

   8. Simplified 29.9%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

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

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 28.8

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

   11. Simplified 28.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 4.8

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

   5. Simplified 4.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 42.2

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

   11. Simplified 42.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in 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 75.5

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

   8. Simplified 75.5%

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

4. Final simplification 58.4%

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

Alternative 8: 56.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (*
      (* re (* re re))
      (fma (* im im) -0.08333333333333333 0.16666666666666666))
     (if (<= t_0 0.0)
       (* 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 <= -0.05) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, 0.16666666666666666);
	} else if (t_0 <= 0.0) {
		tmp = 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 <= -0.05)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666));
	elseif (t_0 <= 0.0)
		tmp = 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, -0.05], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $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)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-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 31.7

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

   5. Simplified 31.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 29.9

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

   8. Simplified 29.9%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. *-commutative N/A

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

       11. +-commutative N/A

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

       12. distribute-lft1-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + \frac{1}{6}\right) \]

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

       20. metadata-eval 28.3

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right) \]

   11. Simplified 28.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 4.8

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

   5. Simplified 4.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 42.2

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

   11. Simplified 42.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in 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 75.5

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

   8. Simplified 75.5%

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

Alternative 9: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.02)
     (* (+ re 1.0) (fma im (* im -0.5) 1.0))
     (if (<= t_0 0.0)
       (* 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 <= -0.02) {
		tmp = (re + 1.0) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 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 <= -0.02)
		tmp = Float64(Float64(re + 1.0) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 0.0)
		tmp = 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, -0.02], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $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)) < -0.0200000000000000004

   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 31.1

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

   5. Simplified 31.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 22.3

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

   8. Simplified 22.3%

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

   if -0.0200000000000000004 < (*.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

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

      1. cos-lowering-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 42.9

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

   11. Simplified 42.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in 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 75.5

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

   8. Simplified 75.5%

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

Alternative 10: 41.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (fma -0.5 (* im im) 1.0)
     (if (<= t_0 2.0) (+ re 1.0) (* re (fma re 0.5 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * fma(re, 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * fma(re, 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, 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 35.8

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

   5. Simplified 35.8%

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

   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 10.2

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

   8. Simplified 10.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 74.4

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

   5. Simplified 74.4%

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

   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 73.1

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 73.1%

      \[\leadsto \color{blue}{re + 1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 98.3

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

   5. Simplified 98.3%

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

   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 65.3

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

   8. Simplified 65.3%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. distribute-rgt-in N/A

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

       4. lft-mult-inverse N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 65.3

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

   11. Simplified 65.3%

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

Alternative 11: 45.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (+ re 1.0) (fma im (* im -0.5) 1.0))
   (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)) <= 0.0) {
		tmp = (re + 1.0) * fma(im, (im * -0.5), 1.0);
	} 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)) <= 0.0)
		tmp = Float64(Float64(re + 1.0) * fma(im, Float64(im * -0.5), 1.0));
	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], 0.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $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)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-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 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 11.7

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

   8. Simplified 11.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in 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 75.5

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

   8. Simplified 75.5%

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

Alternative 12: 44.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma -0.5 (* im im) 1.0)
   (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)) <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} 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)) <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	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], 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $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)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 35.8

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

   5. Simplified 35.8%

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

   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 10.2

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

   8. Simplified 10.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in 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 75.5

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

   8. Simplified 75.5%

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

Alternative 13: 41.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (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)) <= 0.0) {
		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)) <= 0.0)
		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], 0.0], 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)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 35.8

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

   5. Simplified 35.8%

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

   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 10.2

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

   8. Simplified 10.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 82.0

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

   5. Simplified 82.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. 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 70.8

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

   8. Simplified 70.8%

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

Alternative 14: 37.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2.0) (+ re 1.0) (* re (fma re 0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * fma(re, 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * fma(re, 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 2.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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 63.0

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

   5. Simplified 63.0%

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

   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 35.4

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 35.4%

      \[\leadsto \color{blue}{re + 1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 98.3

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

   5. Simplified 98.3%

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

   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 65.3

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

   8. Simplified 65.3%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. distribute-rgt-in N/A

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

       4. lft-mult-inverse N/A

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

       5. +-commutative N/A

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

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

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 65.3

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

   11. Simplified 65.3%

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

Alternative 15: 37.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2.0) (+ re 1.0) (* 0.5 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 2.0d0) then
        tmp = re + 1.0d0
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 2.0:
		tmp = re + 1.0
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 2.0)
		tmp = re + 1.0;
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 2.0], N[(re + 1.0), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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 63.0

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

   5. Simplified 63.0%

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

   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 35.4

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 35.4%

      \[\leadsto \color{blue}{re + 1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 98.3

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

   5. Simplified 98.3%

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

   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 65.3

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

   8. Simplified 65.3%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 65.3

          \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot re\right)} \]

   11. Simplified 65.3%

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

Alternative 16: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.155:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0042)
   (exp re)
   (if (<= re 0.155)
     (* (cos im) (fma (fma re 0.16666666666666666 0.5) (* re re) (+ re 1.0)))
     (if (<= re 2.05e+102)
       (* (exp re) (fma im (* im -0.5) 1.0))
       (*
        (cos im)
        (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0042) {
		tmp = exp(re);
	} else if (re <= 0.155) {
		tmp = cos(im) * fma(fma(re, 0.16666666666666666, 0.5), (re * re), (re + 1.0));
	} else if (re <= 2.05e+102) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else {
		tmp = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0042)
		tmp = exp(re);
	elseif (re <= 0.155)
		tmp = Float64(cos(im) * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), Float64(re + 1.0)));
	elseif (re <= 2.05e+102)
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	else
		tmp = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0042], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.155], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.05e+102], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.00419999999999999974

   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 100.0

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

   5. Simplified 100.0%

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

   if -0.00419999999999999974 < re < 0.154999999999999999

   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. 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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 0.154999999999999999 < re < 2.05e102

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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 85.0

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

   5. Simplified 85.0%

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

   if 2.05e102 < 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%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.155:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re + 1\right)\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.155:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))
   (if (<= re -0.0042)
     (exp re)
     (if (<= re 0.155)
       t_0
       (if (<= re 2.05e+102) (* (exp re) (fma im (* im -0.5) 1.0)) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (re <= -0.0042) {
		tmp = exp(re);
	} else if (re <= 0.155) {
		tmp = t_0;
	} else if (re <= 2.05e+102) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	tmp = 0.0
	if (re <= -0.0042)
		tmp = exp(re);
	elseif (re <= 0.155)
		tmp = t_0;
	elseif (re <= 2.05e+102)
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0042], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.155], t$95$0, If[LessEqual[re, 2.05e+102], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00419999999999999974

   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 100.0

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

   5. Simplified 100.0%

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

   if -0.00419999999999999974 < re < 0.154999999999999999 or 2.05e102 < 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%

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

   if 0.154999999999999999 < re < 2.05e102

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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 85.0

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

   5. Simplified 85.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.155:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;re \leq -1.25 \cdot 10^{+18}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.00034:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(re, t\_0 \cdot \left(re \cdot t\_0\right), -1\right)}{\mathsf{fma}\left(re, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re 0.16666666666666666 0.5) 1.0)))
   (if (<= re -1.25e+18)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 0.00034)
       (cos im)
       (if (<= re 2.05e+102)
         (/
          (* (fma im (* im -0.5) 1.0) (fma re (* t_0 (* re t_0)) -1.0))
          (fma re t_0 -1.0))
         (fma re t_0 1.0))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (re <= -1.25e+18) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 0.00034) {
		tmp = cos(im);
	} else if (re <= 2.05e+102) {
		tmp = (fma(im, (im * -0.5), 1.0) * fma(re, (t_0 * (re * t_0)), -1.0)) / fma(re, t_0, -1.0);
	} else {
		tmp = fma(re, t_0, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (re <= -1.25e+18)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 0.00034)
		tmp = cos(im);
	elseif (re <= 2.05e+102)
		tmp = Float64(Float64(fma(im, Float64(im * -0.5), 1.0) * fma(re, Float64(t_0 * Float64(re * t_0)), -1.0)) / fma(re, t_0, -1.0));
	else
		tmp = fma(re, t_0, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[re, -1.25e+18], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00034], N[Cos[im], $MachinePrecision], If[LessEqual[re, 2.05e+102], N[(N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(t$95$0 * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(re * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(re * t$95$0 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.25e18

   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 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 44.4

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

   11. Simplified 44.4%

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

   if -1.25e18 < re < 3.4e-4

   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 96.9

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

   5. Simplified 96.9%

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

   if 3.4e-4 < re < 2.05e102

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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 85.0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 44.9

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

   8. Simplified 44.9%

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

      1. flip-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 77.4%

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

   if 2.05e102 < 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 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 85.7

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

   8. Simplified 85.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{+18}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.00034:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right) \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right), -1\right)}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.4% 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 69.3

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

5. Simplified 69.3%

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

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 30.1

      \[\leadsto \color{blue}{re + 1} \]

8. Simplified 30.1%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 20: 27.9% 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 53.7

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

5. Simplified 53.7%

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

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 29.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))