math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 22

Speedup: 1.0×

• 25.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot e^{re}\\ t_1 := \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re, -1\right), \frac{1}{re - 1}, \left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (exp re))) (t_1 (* (* (* im im) -0.5) (exp re))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -0.04)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 0.9999)
           (*
            (fma
             (fma re re -1.0)
             (/ 1.0 (- re 1.0))
             (* (* (fma 0.16666666666666666 re 0.5) re) re))
            (cos im))
           (*
            (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)
            (exp re))))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * exp(re);
	double t_1 = ((im * im) * -0.5) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -0.04) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 0.9999) {
		tmp = fma(fma(re, re, -1.0), (1.0 / (re - 1.0)), ((fma(0.16666666666666666, re, 0.5) * re) * re)) * cos(im);
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0) * exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * exp(re))
	t_1 = Float64(Float64(Float64(im * im) * -0.5) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -0.04)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 0.9999)
		tmp = Float64(fma(fma(re, re, -1.0), Float64(1.0 / Float64(re - 1.0)), Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re)) * cos(im));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0) * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -0.04], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 0.9999], N[(N[(N[(re * re + -1.0), $MachinePrecision] * N[(1.0 / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0 or -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 83.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0.9999:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re, -1\right), \frac{1}{re - 1}, \left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot e^{re}\\ t_1 := \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9999998:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (exp re))) (t_1 (* (* (* im im) -0.5) (exp re))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -0.04)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 0.9999998)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (cos im))
           (*
            (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0)
            (exp re))))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * exp(re);
	double t_1 = ((im * im) * -0.5) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -0.04) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 0.9999998) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = fma((0.041666666666666664 * (im * im)), (im * im), 1.0) * exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * exp(re))
	t_1 = Float64(Float64(Float64(im * im) * -0.5) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -0.04)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 0.9999998)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	else
		tmp = Float64(fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0) * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -0.04], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 0.9999998], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0 or -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 83.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999799999999994

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.999999799999999994 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0.9999998:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + re\right) \cdot \cos im\\ t_1 := \cos im \cdot e^{re}\\ t_2 := \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.9999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 re) (cos im)))
        (t_1 (* (cos im) (exp re)))
        (t_2 (* (* (* im im) -0.5) (exp re))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -0.04)
       t_0
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 0.9999)
           t_0
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = (1.0 + re) * cos(im);
	double t_1 = cos(im) * exp(re);
	double t_2 = ((im * im) * -0.5) * exp(re);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -0.04) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 0.9999) {
		tmp = t_0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(1.0 + re) * cos(im))
	t_1 = Float64(cos(im) * exp(re))
	t_2 = Float64(Float64(Float64(im * im) * -0.5) * exp(re))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -0.04)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 0.9999)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -0.04], t$95$0, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 0.9999], t$95$0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0 or -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 83.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 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

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

      1. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.0

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

   8. Applied rewrites 88.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.5\right) \cdot e^{re}\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0.9999:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot e^{re}\\ t_1 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({im}^{4} \cdot 0.041666666666666664\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (exp re))) (t_1 (* (+ 1.0 re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0)
      (fma (* (* re re) 0.16666666666666666) re 1.0))
     (if (<= t_0 -0.04)
       t_1
       (if (<= t_0 0.0)
         (* (* (pow im 4.0) 0.041666666666666664) (+ 1.0 re))
         (if (<= t_0 0.9999)
           t_1
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * exp(re);
	double t_1 = (1.0 + re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0) * fma(((re * re) * 0.16666666666666666), re, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (pow(im, 4.0) * 0.041666666666666664) * (1.0 + re);
	} else if (t_0 <= 0.9999) {
		tmp = t_1;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * exp(re))
	t_1 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0) * fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0));
	elseif (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((im ^ 4.0) * 0.041666666666666664) * Float64(1.0 + re));
	elseif (t_0 <= 0.9999)
		tmp = t_1;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999], t$95$1, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot {re}^{2}, re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 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

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

      1. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 31.1%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]

   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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.0

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

   8. Applied rewrites 88.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0:\\ \;\;\;\;\left({im}^{4} \cdot 0.041666666666666664\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0.9999:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot e^{re}\\ t_1 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (exp re))) (t_1 (* (+ 1.0 re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0)
      (fma (* (* re re) 0.16666666666666666) re 1.0))
     (if (<= t_0 -0.04)
       t_1
       (if (<= t_0 0.0)
         (* (* -0.5 im) im)
         (if (<= t_0 0.9999)
           t_1
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * exp(re);
	double t_1 = (1.0 + re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0) * fma(((re * re) * 0.16666666666666666), re, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else if (t_0 <= 0.9999) {
		tmp = t_1;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * exp(re))
	t_1 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0) * fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0));
	elseif (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	elseif (t_0 <= 0.9999)
		tmp = t_1;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.9999], t$95$1, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot {re}^{2}, re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 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

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

      1. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.0400000000000000008 < (*.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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.0

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

   8. Applied rewrites 88.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0.9999:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (exp re))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0)
      (fma (* (* re re) 0.16666666666666666) re 1.0))
     (if (<= t_0 -0.04)
       (cos im)
       (if (<= t_0 0.0)
         (* (* -0.5 im) im)
         (if (<= t_0 0.9999)
           (cos im)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0) * fma(((re * re) * 0.16666666666666666), re, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else if (t_0 <= 0.9999) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0) * fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0));
	elseif (t_0 <= -0.04)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	elseif (t_0 <= 0.9999)
		tmp = cos(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.9999], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot {re}^{2}, re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 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 96.7

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

   5. Applied rewrites 96.7%

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

   if -0.0400000000000000008 < (*.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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.0

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

   8. Applied rewrites 88.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;\cos im \cdot e^{re} \leq 0.9999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $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 33.2

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

   5. Applied rewrites 33.2%

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

   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 10.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 24.1%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.6

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

   8. Applied rewrites 69.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

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

Alternative 9: 48.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $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 33.2

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

   5. Applied rewrites 33.2%

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

   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 10.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 24.1%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.3

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

   8. Applied rewrites 64.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

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

Alternative 10: 48.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $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 33.2

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

   5. Applied rewrites 33.2%

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

   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 10.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 24.1%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.3

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

   8. Applied rewrites 64.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 64.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), \color{blue}{im} \cdot im, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

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

Alternative 11: 46.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \leq 1.001:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)))
   (if (<= (exp re) 0.0)
     (* (* -0.5 im) im)
     (if (<= (exp re) 1.001)
       (* t_0 (fma (fma 0.5 re 1.0) re 1.0))
       (* (* (* (fma 0.16666666666666666 re 0.5) re) re) t_0)))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else if (exp(re) <= 1.001) {
		tmp = t_0 * fma(fma(0.5, re, 1.0), re, 1.0);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	elseif (exp(re) <= 1.001)
		tmp = Float64(t_0 * fma(fma(0.5, re, 1.0), re, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.001], N[(t$95$0 * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (exp.f64 re) < 1.0009999999999999

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.5

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

   8. Applied rewrites 50.5%

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

   if 1.0009999999999999 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 58.7

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

   8. Applied rewrites 58.7%

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

   10. Applied rewrites 58.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right)}{\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 58.7%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \leq 1.001:\\ \;\;\;\;\left(1 + re\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)))
   (if (<= (exp re) 0.0)
     (* (* -0.5 im) im)
     (if (<= (exp re) 1.001)
       (* (+ 1.0 re) t_0)
       (* (* (fma 0.5 re 1.0) re) t_0)))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else if (exp(re) <= 1.001) {
		tmp = (1.0 + re) * t_0;
	} else {
		tmp = (fma(0.5, re, 1.0) * re) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	elseif (exp(re) <= 1.001)
		tmp = Float64(Float64(1.0 + re) * t_0);
	else
		tmp = Float64(Float64(fma(0.5, re, 1.0) * re) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.001], N[(N[(1.0 + re), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (exp.f64 re) < 1.0009999999999999

   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. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.9

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

   8. Applied rewrites 49.9%

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

   if 1.0009999999999999 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 48.1%

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

Alternative 13: 41.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Cos[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $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 33.2

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

   5. Applied rewrites 33.2%

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

   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 10.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 24.1%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 63.6

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

   5. Applied rewrites 63.6%

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

   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 49.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.8%

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

Alternative 14: 46.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.3

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

   8. Applied rewrites 53.3%

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

4. Final simplification 47.0%

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

Alternative 15: 46.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.3

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 52.0%

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

Alternative 16: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{if}\;re \leq -0.012:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (* im im) -0.5 1.0) (exp re))))
   (if (<= re -0.012)
     t_0
     (if (<= re 0.023)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
       (if (<= re 1.9e+154) t_0 (* (fma (fma 0.5 re 1.0) re 1.0) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0) * exp(re);
	double tmp;
	if (re <= -0.012) {
		tmp = t_0;
	} else if (re <= 0.023) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(Float64(im * im), -0.5, 1.0) * exp(re))
	tmp = 0.0
	if (re <= -0.012)
		tmp = t_0;
	elseif (re <= 0.023)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.012], t$95$0, If[LessEqual[re, 0.023], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], t$95$0, N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.012 or 0.023 < re < 1.8999999999999999e154

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -0.012 < re < 0.023

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.012:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 90.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ t_1 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{if}\;re \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 0.0044:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (fma 0.5 re 1.0) re 1.0) (cos im)))
        (t_1 (* (fma (* im im) -0.5 1.0) (exp re))))
   (if (<= re -1.42e-5)
     t_1
     (if (<= re 0.0044) t_0 (if (<= re 1.9e+154) t_1 t_0)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	double t_1 = fma((im * im), -0.5, 1.0) * exp(re);
	double tmp;
	if (re <= -1.42e-5) {
		tmp = t_1;
	} else if (re <= 0.0044) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im))
	t_1 = Float64(fma(Float64(im * im), -0.5, 1.0) * exp(re))
	tmp = 0.0
	if (re <= -1.42e-5)
		tmp = t_1;
	elseif (re <= 0.0044)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.42e-5], t$95$1, If[LessEqual[re, 0.0044], t$95$0, If[LessEqual[re, 1.9e+154], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if re < -1.42e-5 or 0.00440000000000000027 < re < 1.8999999999999999e154

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -1.42e-5 < re < 0.00440000000000000027 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.0044:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.7

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

   8. Applied rewrites 49.7%

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

4. Final simplification 44.3%

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

Alternative 19: 86.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{if}\;re \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.0017:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (* im im) -0.5 1.0) (exp re))))
   (if (<= re -2.7e-6) t_0 (if (<= re 0.0017) (* (+ 1.0 re) (cos im)) t_0))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0) * exp(re);
	double tmp;
	if (re <= -2.7e-6) {
		tmp = t_0;
	} else if (re <= 0.0017) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(Float64(im * im), -0.5, 1.0) * exp(re))
	tmp = 0.0
	if (re <= -2.7e-6)
		tmp = t_0;
	elseif (re <= 0.0017)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.7e-6], t$95$0, If[LessEqual[re, 0.0017], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -2.69999999999999998e-6 or 0.00169999999999999991 < re

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -2.69999999999999998e-6 < re < 0.00169999999999999991

   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. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.0017:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.1

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

   8. Applied rewrites 39.1%

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

Alternative 21: 35.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 2.3e-183) (* (* -0.5 im) im) (fma (* im im) -0.5 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 2.3e-183) {
		tmp = (-0.5 * im) * im;
	} else {
		tmp = fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 2.3e-183)
		tmp = Float64(Float64(-0.5 * im) * im);
	else
		tmp = fma(Float64(im * im), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 2.3e-183], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 2.30000000000000016e-183

   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

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 26.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 2.30000000000000016e-183 < (exp.f64 re)

   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 65.9

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

   5. Applied rewrites 65.9%

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

   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 37.3%

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

Alternative 22: 11.7% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \left(-0.5 \cdot im\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* -0.5 im) im))

C

double code(double re, double im) {
	return (-0.5 * im) * im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-0.5d0) * im) * im
end function

Java

public static double code(double re, double im) {
	return (-0.5 * im) * im;
}

Python

def code(re, im):
	return (-0.5 * im) * im

Julia

function code(re, im)
	return Float64(Float64(-0.5 * im) * im)
end

MATLAB

function tmp = code(re, im)
	tmp = (-0.5 * im) * im;
end

Wolfram

code[re_, im_] := N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. lower-cos.f64 50.9

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

5. Applied rewrites 50.9%

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

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 29.0%

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

9. Taylor expanded in im around inf

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

11. Applied rewrites 11.0%

    \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
12. Step-by-step derivation

13. Applied rewrites 11.0%

    \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))