math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 20

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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (fma re (fma re 0.5 1.0) 1.0)))
        (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (fma im (* im -0.5) 1.0))
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 0.0)
         (exp re)
         (if (<= t_1 0.9999999999999999) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999999999999) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999999999999)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.999999999999999889 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999999999:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ re 1.0))) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (fma im (* im -0.5) 1.0))
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 0.0)
         (exp re)
         (if (<= t_1 0.9999999999999999) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (re + 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999999999999) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(re + 1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999999999999)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.999999999999999889 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999999999:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (cos im) (+ re 1.0))))
   (if (<= t_0 (- INFINITY))
     (fma
      (* im im)
      (fma
       im
       (* im (fma (* im im) -0.001388888888888889 0.041666666666666664))
       -0.5)
      1.0)
     (if (<= t_0 -0.02)
       t_1
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 0.9999999999999999) t_1 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * (re + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((im * im), fma(im, (im * fma((im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999999999999)
		tmp = t_1;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], t$95$1, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 93.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.999999999999999889 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999999999:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (* im im)
      (fma
       im
       (* im (fma (* im im) -0.001388888888888889 0.041666666666666664))
       -0.5)
      1.0)
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 0.9999999999999999) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((im * im), fma(im, (im * fma((im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999999999999)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 93.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999889

   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 97.1

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

   5. Applied rewrites 97.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.999999999999999889 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 76.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (* im im)
      (fma
       im
       (* im (fma (* im im) -0.001388888888888889 0.041666666666666664))
       -0.5)
      1.0)
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (* (* im im) (* im (* im 0.041666666666666664)))
         (if (<= t_0 0.9999)
           (cos im)
           (*
            (fma re (fma re 0.5 1.0) 1.0)
            (fma
             im
             (* im (fma im (* im 0.041666666666666664) -0.5))
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((im * im), fma(im, (im * fma((im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (im * im) * (im * (im * 0.041666666666666664));
	} else if (t_0 <= 0.9999) {
		tmp = cos(im);
	} else {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, (im * fma(im, (im * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664)));
	elseif (t_0 <= 0.9999)
		tmp = cos(im);
	else
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), -0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999], N[Cos[im], $MachinePrecision], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 93.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99990000000000001

   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 96.9

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

   5. Applied rewrites 96.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 48.5%

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

   if 0.99990000000000001 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 77.9

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 90.6

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

   11. Applied rewrites 90.6%

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

Alternative 7: 55.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(1 + \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.02)
     (fma
      (* im im)
      (fma
       im
       (* im (fma (* im im) -0.001388888888888889 0.041666666666666664))
       -0.5)
      1.0)
     (if (<= t_0 0.0)
       (* (* im im) (* im (* im 0.041666666666666664)))
       (+ re (+ 1.0 (* (fma re 0.16666666666666666 0.5) (* re re))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma((im * im), fma(im, (im * fma((im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (im * im) * (im * (im * 0.041666666666666664));
	} else {
		tmp = re + (1.0 + (fma(re, 0.16666666666666666, 0.5) * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664)));
	else
		tmp = Float64(re + Float64(1.0 + Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(1.0 + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 32.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 48.5%

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

   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   10. Applied rewrites 69.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(1 + \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664)));
	else
		tmp = Float64(re + Float64(1.0 + Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(1.0 + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 34.9

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

   5. Applied rewrites 34.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 30.8

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

   8. Applied rewrites 30.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 48.5%

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

   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   10. Applied rewrites 69.2%

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

4. Final simplification 58.0%

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

Alternative 9: 56.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(1 + \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (* (fma im (* im -0.5) 1.0) (* re (* re 0.5)))
     (if (<= t_0 0.0)
       (* (* im im) (* im (* im 0.041666666666666664)))
       (+ re (+ 1.0 (* (fma re 0.16666666666666666 0.5) (* re re))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(im, (im * -0.5), 1.0) * (re * (re * 0.5));
	} else if (t_0 <= 0.0) {
		tmp = (im * im) * (im * (im * 0.041666666666666664));
	} else {
		tmp = re + (1.0 + (fma(re, 0.16666666666666666, 0.5) * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * Float64(re * Float64(re * 0.5)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664)));
	else
		tmp = Float64(re + Float64(1.0 + Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(1.0 + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 36.4

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 32.1

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

   8. Applied rewrites 32.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 32.2%

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

   if -0.10000000000000001 < (*.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 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 46.8%

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

   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   10. Applied rewrites 69.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + \left(1 + \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * Float64(re + 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664)));
	else
		tmp = Float64(re + Float64(1.0 + Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + N[(1.0 + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 34.9

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

   5. Applied rewrites 34.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 30.6

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

   8. Applied rewrites 30.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 48.5%

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

   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   10. Applied rewrites 69.2%

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

4. Final simplification 57.9%

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

Alternative 11: 55.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * Float64(re + 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * 0.041666666666666664)));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 34.9

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

   5. Applied rewrites 34.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 30.6

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

   8. Applied rewrites 30.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 48.5%

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

   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

4. Final simplification 57.9%

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

Alternative 12: 51.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* im (* im -0.5))
     (if (<= t_0 2.0)
       (fma re (fma re 0.5 1.0) 1.0)
       (* (fma re 0.16666666666666666 0.5) (* re re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = im * (im * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = fma(re, 0.16666666666666666, 0.5) * (re * re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (t_0 <= 2.0)
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	else
		tmp = Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 74.7%

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

Alternative 13: 51.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;im \cdot \left(im \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* im (* im -0.5))
     (if (<= t_0 2.0)
       (fma re (fma re 0.5 1.0) 1.0)
       (* 0.16666666666666666 (* re (* re re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = im * (im * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = 0.16666666666666666 * (re * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(im * Float64(im * -0.5));
	elseif (t_0 <= 2.0)
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	else
		tmp = Float64(0.16666666666666666 * Float64(re * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 74.7%

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

Alternative 14: 48.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* im (* im -0.5))
     (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 = im * (im * -0.5);
	} 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 = Float64(im * Float64(im * -0.5));
	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[(im * N[(im * -0.5), $MachinePrecision]), $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 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 74.7%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 65.1%

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

Alternative 15: 51.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* im (* im -0.5))
   (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);
	} 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(im * Float64(im * -0.5));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -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 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

Alternative 16: 50.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -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 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 68.8%

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

Alternative 17: 48.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.1%

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

Alternative 18: 38.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 0.0d0) then
        tmp = im * (im * (-0.5d0))
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 0.0:
		tmp = im * (im * -0.5)
	else:
		tmp = re + 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -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 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.2%

      \[\leadsto re + \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 29.4% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-exp.f64 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.3%

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

Alternative 20: 28.9% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-exp.f64 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 28.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))