math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 22.5s

Alternatives: 17

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(e^{re} \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999435:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (* im im) (* (exp re) -0.5))
     (if (<= t_0 -0.05)
       (cos im)
       (if (<= t_0 0.0) (exp re) (if (<= t_0 0.9999435) (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 = (im * im) * (exp(re) * -0.5);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999435) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (im * im) * (Math.exp(re) * -0.5);
	} else if (t_0 <= -0.05) {
		tmp = Math.cos(im);
	} else if (t_0 <= 0.0) {
		tmp = Math.exp(re);
	} else if (t_0 <= 0.9999435) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (im * im) * (math.exp(re) * -0.5)
	elif t_0 <= -0.05:
		tmp = math.cos(im)
	elif t_0 <= 0.0:
		tmp = math.exp(re)
	elif t_0 <= 0.9999435:
		tmp = math.cos(im)
	else:
		tmp = math.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(im * im) * Float64(exp(re) * -0.5));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999435)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (im * im) * (exp(re) * -0.5);
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999435)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(N[Exp[re], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999435], 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. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999943499999999985

   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. lower-cos.f64 100.0

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

   5. Simplified 100.0%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999943499999999985 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(e^{re} \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999435:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \left(re \cdot \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot -0.5, re\right)\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999435:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma (* re re) 0.027777777777777776 -0.25)
       (* re (fma re (* (* im im) -0.5) re)))
      (fma re 0.16666666666666666 -0.5))
     (if (<= t_0 -0.05)
       (cos im)
       (if (<= t_0 0.0) (exp re) (if (<= t_0 0.9999435) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((re * re), 0.027777777777777776, -0.25) * (re * fma(re, ((im * im) * -0.5), re))) / fma(re, 0.16666666666666666, -0.5);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999435) {
		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(fma(Float64(re * re), 0.027777777777777776, -0.25) * Float64(re * fma(re, Float64(Float64(im * im) * -0.5), re))) / fma(re, 0.16666666666666666, -0.5));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999435)
		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[(N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision] * N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999435], 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. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. 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. lower-fma.f64 N/A

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

      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. lower-fma.f64 N/A

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

      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. lower-fma.f64 78.9

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

   8. Simplified 78.9%

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

   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. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 78.9

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

   11. Simplified 78.9%

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

   13. Applied egg-rr 85.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \left(re \cdot \mathsf{fma}\left(re, -0.5 \cdot \left(im \cdot im\right), re\right)\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999943499999999985

   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. lower-cos.f64 100.0

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

   5. Simplified 100.0%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999943499999999985 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \left(re \cdot \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot -0.5, re\right)\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999435:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* -0.5 (* im (fma re im im)))
     (if (<= t_0 20.0)
       (fma re (fma 0.5 re 1.0) 1.0)
       (* (* re re) (fma re 0.16666666666666666 0.5))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * (im * fma(re, im, im));
	} else if (t_0 <= 20.0) {
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	} else {
		tmp = (re * re) * fma(re, 0.16666666666666666, 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(im * fma(re, im, im)));
	elseif (t_0 <= 20.0)
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	else
		tmp = Float64(Float64(re * re) * fma(re, 0.16666666666666666, 0.5));
	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 * N[(re * im + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 20.0], N[(re * N[(0.5 * re + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in re around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 8.5

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

   8. Simplified 8.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 19.3

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

   11. Simplified 19.3%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 69.4

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

   8. Simplified 69.4%

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

   if 20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 65.9

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

   8. Simplified 65.9%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 65.9

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

   11. Simplified 65.9%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 20:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* -0.5 (* im (fma re im im)))
     (if (<= t_0 20.0)
       (fma re (fma 0.5 re 1.0) 1.0)
       (* re (* re (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * (im * fma(re, im, im));
	} else if (t_0 <= 20.0) {
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	} else {
		tmp = re * (re * (re * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(im * fma(re, im, im)));
	elseif (t_0 <= 20.0)
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	else
		tmp = Float64(re * Float64(re * Float64(re * 0.16666666666666666)));
	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 * N[(re * im + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 20.0], N[(re * N[(0.5 * re + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in re around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 8.5

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

   8. Simplified 8.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 19.3

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

   11. Simplified 19.3%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 69.4

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

   8. Simplified 69.4%

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

   if 20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 65.9

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

   8. Simplified 65.9%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \frac{1}{6}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 65.9

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

   11. Simplified 65.9%

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

Alternative 6: 44.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

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 20.0)
       (fma re (fma 0.5 re 1.0) 1.0)
       (* re (* re (* re 0.16666666666666666)))))))

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 <= 20.0) {
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	} else {
		tmp = re * (re * (re * 0.16666666666666666));
	}
	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 <= 20.0)
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	else
		tmp = Float64(re * Float64(re * Float64(re * 0.16666666666666666)));
	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, 20.0], N[(re * N[(0.5 * re + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 28.3

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

   5. Simplified 28.3%

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

   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. lower-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. lower-*.f64 7.9

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

   8. Simplified 7.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 69.4

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

   8. Simplified 69.4%

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

   if 20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 65.9

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

   8. Simplified 65.9%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \frac{1}{6}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 65.9

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

   11. Simplified 65.9%

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

Alternative 7: 41.1% accurate, 0.5× speedup

Math

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

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) (fma re (* re 0.5) re)))))

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 = fma(re, (re * 0.5), re);
	}
	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 = fma(re, Float64(re * 0.5), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(-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), $MachinePrecision] + re), $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. lower-cos.f64 28.3

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

   5. Simplified 28.3%

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

   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. lower-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. lower-*.f64 7.9

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

   8. Simplified 7.9%

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

   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. lower-exp.f64 70.7

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

   5. Simplified 70.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 69.5

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

   8. Simplified 69.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 46.4

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

   8. Simplified 46.4%

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

   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. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 46.4

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

   11. Simplified 46.4%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot 0.5, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

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 20.0) (+ re 1.0) (* (* re re) 0.5)))))

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 <= 20.0) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	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 <= 20.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	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, 20.0], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $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. lower-cos.f64 28.3

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

   5. Simplified 28.3%

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

   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. lower-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. lower-*.f64 7.9

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

   8. Simplified 7.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 69.0

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

   8. Simplified 69.0%

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

   if 20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 46.9

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

   8. Simplified 46.9%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.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. lower-*.f64 46.9

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

   11. Simplified 46.9%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 20:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(re, 0.16666666666666666, 0.5) * (-0.5 * (re * (re * (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)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(fma(re, 0.16666666666666666, 0.5) * Float64(-0.5 * Float64(re * Float64(re * 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_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(-0.5 * N[(re * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   5. Simplified 62.6%

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

   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. lower-fma.f64 N/A

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

      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. lower-fma.f64 N/A

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

      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. lower-fma.f64 11.9

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

   8. Simplified 11.9%

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

   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. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 11.7

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

   11. Simplified 11.7%

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

   12. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 28.1

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

   14. Simplified 28.1%

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

   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 im around 0

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

      1. lower-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 68.4

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

   8. Simplified 68.4%

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

Alternative 10: 51.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (* (* im im) -0.5) (+ re 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 = ((im * im) * -0.5) * (re + 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(Float64(im * im) * -0.5) * Float64(re + 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[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[(re + 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. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in re around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 8.5

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

   8. Simplified 8.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 24.7

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

   11. Simplified 24.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 68.4

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

   8. Simplified 68.4%

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

4. Final simplification 50.6%

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

Alternative 11: 51.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in re around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 8.5

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

   8. Simplified 8.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 24.7

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

   11. Simplified 24.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 68.4

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

   8. Simplified 68.4%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. lift-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 68.4

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 68.4

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

   10. Applied egg-rr 68.4%

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

   11. Taylor expanded in re around 0

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

   13. Simplified 68.0%

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

4. Final simplification 50.4%

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

Alternative 12: 48.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in re around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 8.5

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

   8. Simplified 8.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 19.3

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

   11. Simplified 19.3%

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

   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 im around 0

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

      1. lower-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 68.4

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

   8. Simplified 68.4%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. lift-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 68.4

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 68.4

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

   10. Applied egg-rr 68.4%

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

   11. Taylor expanded in re around 0

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

   13. Simplified 68.0%

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

Alternative 13: 41.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma -0.5 (* im im) 1.0)
   (fma re (fma 0.5 re 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(0.5, re, 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(0.5, re, 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[(0.5 * re + 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. lower-cos.f64 28.3

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

   5. Simplified 28.3%

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

   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. lower-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. lower-*.f64 7.9

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

   8. Simplified 7.9%

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

   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. lower-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 62.0

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

   8. Simplified 62.0%

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

Alternative 14: 37.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 20:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 20.0) (+ re 1.0) (* (* re re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 20.0) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	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)) <= 20.0d0) then
        tmp = re + 1.0d0
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 20.0) {
		tmp = re + 1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 20.0:
		tmp = re + 1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 20.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 20.0)
		tmp = re + 1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 20.0], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 35.1

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

   8. Simplified 35.1%

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

   if 20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 46.9

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

   8. Simplified 46.9%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.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. lower-*.f64 46.9

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

   11. Simplified 46.9%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 20:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \left(\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, -0.125\right)} \cdot \left(\mathsf{fma}\left(re, re \cdot 0.027777777777777776, 0.25\right) - re \cdot -0.08333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;re \leq -600:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\\ \mathbf{elif}\;re \leq 2.4:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), \left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e+154)
   (* (* (* im im) -0.5) (+ re 1.0))
   (if (<= re -5.6e+102)
     (*
      (*
       (* re re)
       (*
        (/
         (fma (* re re) 0.027777777777777776 -0.25)
         (fma (* re (* re re)) 0.004629629629629629 -0.125))
        (-
         (fma re (* re 0.027777777777777776) 0.25)
         (* re -0.08333333333333333))))
      (fma im (* im -0.5) 1.0))
     (if (<= re -600.0)
       (* -0.5 (* im (fma re im im)))
       (if (<= re 2.4)
         (cos im)
         (if (<= re 7.8e+71)
           (fma
            (* im im)
            (fma
             (* im im)
             (fma im (* im -0.001388888888888889) 0.041666666666666664)
             -0.5)
            1.0)
           (if (<= re 8.2e+148)
             (fma
              (*
               re
               (fma
                (fma re 0.16666666666666666 0.5)
                (* (* re re) (fma re 0.16666666666666666 0.5))
                -1.0))
              (/ 1.0 (fma re (fma re 0.16666666666666666 0.5) -1.0))
              1.0)
             (* (* re re) 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+154) {
		tmp = ((im * im) * -0.5) * (re + 1.0);
	} else if (re <= -5.6e+102) {
		tmp = ((re * re) * ((fma((re * re), 0.027777777777777776, -0.25) / fma((re * (re * re)), 0.004629629629629629, -0.125)) * (fma(re, (re * 0.027777777777777776), 0.25) - (re * -0.08333333333333333)))) * fma(im, (im * -0.5), 1.0);
	} else if (re <= -600.0) {
		tmp = -0.5 * (im * fma(re, im, im));
	} else if (re <= 2.4) {
		tmp = cos(im);
	} else if (re <= 7.8e+71) {
		tmp = fma((im * im), fma((im * im), fma(im, (im * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0);
	} else if (re <= 8.2e+148) {
		tmp = fma((re * fma(fma(re, 0.16666666666666666, 0.5), ((re * re) * fma(re, 0.16666666666666666, 0.5)), -1.0)), (1.0 / fma(re, fma(re, 0.16666666666666666, 0.5), -1.0)), 1.0);
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.35e+154)
		tmp = Float64(Float64(Float64(im * im) * -0.5) * Float64(re + 1.0));
	elseif (re <= -5.6e+102)
		tmp = Float64(Float64(Float64(re * re) * Float64(Float64(fma(Float64(re * re), 0.027777777777777776, -0.25) / fma(Float64(re * Float64(re * re)), 0.004629629629629629, -0.125)) * Float64(fma(re, Float64(re * 0.027777777777777776), 0.25) - Float64(re * -0.08333333333333333)))) * fma(im, Float64(im * -0.5), 1.0));
	elseif (re <= -600.0)
		tmp = Float64(-0.5 * Float64(im * fma(re, im, im)));
	elseif (re <= 2.4)
		tmp = cos(im);
	elseif (re <= 7.8e+71)
		tmp = fma(Float64(im * im), fma(Float64(im * im), fma(im, Float64(im * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0);
	elseif (re <= 8.2e+148)
		tmp = fma(Float64(re * fma(fma(re, 0.16666666666666666, 0.5), Float64(Float64(re * re) * fma(re, 0.16666666666666666, 0.5)), -1.0)), Float64(1.0 / fma(re, fma(re, 0.16666666666666666, 0.5), -1.0)), 1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.35e+154], N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5.6e+102], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision] / N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(re * 0.027777777777777776), $MachinePrecision] + 0.25), $MachinePrecision] - N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -600.0], N[(-0.5 * N[(im * N[(re * im + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.4], N[Cos[im], $MachinePrecision], If[LessEqual[re, 7.8e+71], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[re, 8.2e+148], N[(N[(re * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if re < -1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. 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

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 31.8

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

   11. Simplified 31.8%

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

   if -1.35000000000000003e154 < re < -5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 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. lower-fma.f64 N/A

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

      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. lower-fma.f64 N/A

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

      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. lower-fma.f64 1.6

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

   8. Simplified 1.6%

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

   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. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 1.6

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

   11. Simplified 1.6%

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

   13. Applied egg-rr 85.0%

       \[\leadsto \left(\color{blue}{\left(\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, -0.125\right)} \cdot \left(\mathsf{fma}\left(re, re \cdot 0.027777777777777776, 0.25\right) - re \cdot -0.08333333333333333\right)\right)} \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   if -5.60000000000000037e102 < re < -600

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in re around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 2.2

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

   8. Simplified 2.2%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 26.6

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

   11. Simplified 26.6%

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

   if -600 < re < 2.39999999999999991

   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. lower-cos.f64 98.3

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

   5. Simplified 98.3%

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

   if 2.39999999999999991 < re < 7.8000000000000002e71

   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. lower-cos.f64 3.9

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

   5. Simplified 3.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 33.6

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

   8. Simplified 33.6%

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

   if 7.8000000000000002e71 < re < 8.1999999999999996e148

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 50.5

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

   8. Simplified 50.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. flip-+ N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied egg-rr 85.7%

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

   if 8.1999999999999996e148 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 81.5

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

   8. Simplified 81.5%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.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. lower-*.f64 81.5

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

   11. Simplified 81.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot re\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \left(\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), 0.004629629629629629, -0.125\right)} \cdot \left(\mathsf{fma}\left(re, re \cdot 0.027777777777777776, 0.25\right) - re \cdot -0.08333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;re \leq -600:\\ \;\;\;\;-0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, im, im\right)\right)\\ \mathbf{elif}\;re \leq 2.4:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), \left(re \cdot re\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.9% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-exp.f64 73.0

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

5. Simplified 73.0%

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

6. Taylor expanded in re around 0

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

   1. lower-+.f64 29.2

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

8. Simplified 29.2%

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

9. Final simplification 29.2%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 17: 28.5% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. lower-cos.f64 50.7

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

5. Simplified 50.7%

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

6. Taylor expanded in im around 0

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

8. Simplified 28.7%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))