math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 23

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-\cos im}{\frac{-1}{e^{re}}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (- (cos im)) (/ -1.0 (exp re))))

C

double code(double re, double im) {
	return -cos(im) / (-1.0 / exp(re));
}

Fortran

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

Java

public static double code(double re, double im) {
	return -Math.cos(im) / (-1.0 / Math.exp(re));
}

Python

def code(re, im):
	return -math.cos(im) / (-1.0 / math.exp(re))

Julia

function code(re, im)
	return Float64(Float64(-cos(im)) / Float64(-1.0 / exp(re)))
end

MATLAB

function tmp = code(re, im)
	tmp = -cos(im) / (-1.0 / exp(re));
end

Wolfram

code[re_, im_] := N[((-N[Cos[im], $MachinePrecision]) / N[(-1.0 / N[Exp[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]
5. Add Preprocessing

Alternative 2: 88.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\frac{-\cos im}{re - 1}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (exp re) (* (* im im) -0.5))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -0.05)
       (/ (- (cos im)) (- re 1.0))
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 0.995)
           (* (+ 1.0 re) (cos im))
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (- (* 0.041666666666666664 (* im im)) 0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = exp(re) * ((im * im) * -0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -0.05) {
		tmp = -cos(im) / (re - 1.0);
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 0.995) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(exp(re) * Float64(Float64(im * im) * -0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -0.05)
		tmp = Float64(Float64(-cos(im)) / Float64(re - 1.0));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 0.995)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(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[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -0.05], N[((-N[Cos[im], $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 0.995], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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 or -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 79.7

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

   5. Applied rewrites 79.7%

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

   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 79.7%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 100.0

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.994999999999999996 < (*.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 88.4

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

   5. Applied rewrites 88.4%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 3: 88.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + re\right) \cdot \cos im\\ t_1 := e^{re} \cdot \cos im\\ t_2 := e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 re) (cos im)))
        (t_1 (* (exp re) (cos im)))
        (t_2 (* (exp re) (* (* im im) -0.5))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -0.05)
       t_0
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 0.995)
           t_0
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (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 = exp(re) * cos(im);
	double t_2 = exp(re) * ((im * im) * -0.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 0.995) {
		tmp = t_0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * 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(exp(re) * cos(im))
	t_2 = Float64(exp(re) * Float64(Float64(im * im) * -0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 0.995)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(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[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 0.995], t$95$0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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 -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 79.7

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

   5. Applied rewrites 79.7%

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

   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 79.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.994999999999999996 < (*.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 88.4

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

   5. Applied rewrites 88.4%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 4: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-t\_0}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (* (+ 1.0 re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (* re re) 0.16666666666666666) re 1.0) t_0)
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 0.0)
         (/
          (- t_0)
          (fma (* (- (* 0.16666666666666666 re) 0.5) re) re (- re 1.0)))
         (if (<= t_1 0.995)
           t_2
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (- (* 0.041666666666666664 (* im im)) 0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = (1.0 + re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * t_0;
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = -t_0 / fma((((0.16666666666666666 * re) - 0.5) * re), re, (re - 1.0));
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * t_0);
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-t_0) / fma(Float64(Float64(Float64(0.16666666666666666 * re) - 0.5) * re), re, Float64(re - 1.0)));
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(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[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 0.0], N[((-t$95$0) / N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision] * re), $MachinePrecision] * re + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$2, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.5

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

   8. Applied rewrites 94.5%

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

   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 94.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   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 99.7

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.4

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

   10. Applied rewrites 49.4%

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

   if 0.994999999999999996 < (*.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 88.4

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

   5. Applied rewrites 88.4%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 5: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-t\_0}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (* re re) 0.16666666666666666) re 1.0) t_0)
     (if (<= t_1 -0.05)
       (cos im)
       (if (<= t_1 0.0)
         (/
          (- t_0)
          (fma (* (- (* 0.16666666666666666 re) 0.5) re) re (- re 1.0)))
         (if (<= t_1 0.995)
           (cos im)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (- (* 0.041666666666666664 (* im im)) 0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * t_0;
	} else if (t_1 <= -0.05) {
		tmp = cos(im);
	} else if (t_1 <= 0.0) {
		tmp = -t_0 / fma((((0.16666666666666666 * re) - 0.5) * re), re, (re - 1.0));
	} else if (t_1 <= 0.995) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * t_0);
	elseif (t_1 <= -0.05)
		tmp = cos(im);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-t_0) / fma(Float64(Float64(Float64(0.16666666666666666 * re) - 0.5) * re), re, Float64(re - 1.0)));
	elseif (t_1 <= 0.995)
		tmp = cos(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(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[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-t$95$0) / N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision] * re), $MachinePrecision] * re + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.5

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

   8. Applied rewrites 94.5%

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

   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 94.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 98.4

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.4

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

   10. Applied rewrites 49.4%

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

   if 0.994999999999999996 < (*.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 88.4

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

   5. Applied rewrites 88.4%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

Alternative 6: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-t\_0}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (* re re) 0.16666666666666666) re 1.0) t_0)
     (if (<= t_1 0.0)
       (/
        (- t_0)
        (fma (* (- (* 0.16666666666666666 re) 0.5) re) re (- re 1.0)))
       (if (<= t_1 0.995)
         1.0
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * t_0;
	} else if (t_1 <= 0.0) {
		tmp = -t_0 / fma((((0.16666666666666666 * re) - 0.5) * re), re, (re - 1.0));
	} else if (t_1 <= 0.995) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-t_0) / fma(Float64(Float64(Float64(0.16666666666666666 * re) - 0.5) * re), re, Float64(re - 1.0)));
	elseif (t_1 <= 0.995)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(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[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-t$95$0) / N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision] * re), $MachinePrecision] * re + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], 1.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.5

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

   8. Applied rewrites 94.5%

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

   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 94.5%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 83.2

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.3

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

   10. Applied rewrites 32.3%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 95.7%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto 1 \]

   if 0.994999999999999996 < (*.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 88.4

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

   5. Applied rewrites 88.4%

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

   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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{-\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999998866:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right) \cdot im\right) \cdot im, im \cdot im, 1 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (/ (- (* -0.5 (* im im))) (- re 1.0))
       (if (<= t_0 0.9999999999998866)
         1.0
         (fma
          (* (* (fma 0.041666666666666664 re 0.041666666666666664) im) im)
          (* im im)
          (+ 1.0 re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (t_0 <= 0.9999999999998866) {
		tmp = 1.0;
	} else {
		tmp = fma(((fma(0.041666666666666664, re, 0.041666666666666664) * im) * im), (im * im), (1.0 + re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (t_0 <= 0.9999999999998866)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(fma(0.041666666666666664, re, 0.041666666666666664) * im) * im), Float64(im * im), Float64(1.0 + re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999998866], 1.0, N[(N[(N[(N[(0.041666666666666664 * re + 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.5

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

   8. Applied rewrites 94.5%

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

   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 94.5%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 41.2

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 41.2%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 23.2%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999886646

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 95.8%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.0%

       \[\leadsto 1 \]

   if 0.999999999999886646 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 78.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 78.4%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999998866:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right) \cdot im\right) \cdot im, im \cdot im, 1 + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.865:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999998866:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right) \cdot im\right) \cdot im, im \cdot im, 1 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.865)
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (/ (- (* -0.5 (* im im))) (- re 1.0))
       (if (<= t_0 0.9999999999998866)
         1.0
         (fma
          (* (* (fma 0.041666666666666664 re 0.041666666666666664) im) im)
          (* im im)
          (+ 1.0 re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.865) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (t_0 <= 0.9999999999998866) {
		tmp = 1.0;
	} else {
		tmp = fma(((fma(0.041666666666666664, re, 0.041666666666666664) * im) * im), (im * im), (1.0 + re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.865)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (t_0 <= 0.9999999999998866)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(fma(0.041666666666666664, re, 0.041666666666666664) * im) * im), Float64(im * im), Float64(1.0 + re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.865], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999998866], 1.0, N[(N[(N[(N[(0.041666666666666664 * re + 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.86499999999999999

   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 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if -0.86499999999999999 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 35.1

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 25.2%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999886646

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 95.8%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.0%

       \[\leadsto 1 \]

   if 0.999999999999886646 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 78.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 78.4%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.865:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999998866:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right) \cdot im\right) \cdot im, im \cdot im, 1 + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.865:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5, im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.865)
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (/ (- (* -0.5 (* im im))) (- re 1.0))
       (if (<= t_0 2.0)
         1.0
         (*
          (fma (- (* (* im im) 0.041666666666666664) 0.5) (* im im) 1.0)
          re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.865) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma((((im * im) * 0.041666666666666664) - 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.865)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(fma(Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.865], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.86499999999999999

   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 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if -0.86499999999999999 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 35.1

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 25.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 95.6%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.8%

       \[\leadsto 1 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 43.8%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.865:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5, im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.998:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5, im \cdot im, 1 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.998)
     (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (/ (- (* -0.5 (* im im))) (- re 1.0))
       (if (<= t_0 0.995)
         1.0
         (fma
          (- (* (* im im) 0.041666666666666664) 0.5)
          (* im im)
          (+ 1.0 re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (t_0 <= 0.995) {
		tmp = 1.0;
	} else {
		tmp = fma((((im * im) * 0.041666666666666664) - 0.5), (im * im), (1.0 + re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.998)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (t_0 <= 0.995)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5), Float64(im * im), Float64(1.0 + re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.998], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.995], 1.0, N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.998

   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 69.2

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

   8. Applied rewrites 69.2%

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

   if -0.998 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 40.5

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 40.5%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 23.5%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 95.7%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto 1 \]

   if 0.994999999999999996 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 78.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 76.2%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.998:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5, im \cdot im, 1 + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.998:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.998)
     (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (/ (- (* -0.5 (* im im))) (- re 1.0))
       (if (<= t_0 0.995)
         1.0
         (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (t_0 <= 0.995) {
		tmp = 1.0;
	} else {
		tmp = fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.998)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (t_0 <= 0.995)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(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[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.998], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.995], 1.0, N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.998

   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 69.2

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

   8. Applied rewrites 69.2%

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

   if -0.998 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 40.5

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 40.5%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 23.5%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 95.7%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto 1 \]

   if 0.994999999999999996 < (*.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 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 74.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.998:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;\frac{-\cos im}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 0.995)
       (/
        (- (cos im))
        (fma (* (- (* 0.16666666666666666 re) 0.5) re) re (- re 1.0)))
       (*
        (exp re)
        (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= 0.995) {
		tmp = -cos(im) / fma((((0.16666666666666666 * re) - 0.5) * re), re, (re - 1.0));
	} else {
		tmp = exp(re) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.995)
		tmp = Float64(Float64(-cos(im)) / fma(Float64(Float64(Float64(0.16666666666666666 * re) - 0.5) * re), re, Float64(re - 1.0)));
	else
		tmp = Float64(exp(re) * fma(Float64(Float64(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[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.995], N[((-N[Cos[im], $MachinePrecision]) / N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision] * re), $MachinePrecision] * re + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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

   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 0.994999999999999996 < (*.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. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 13: 41.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (* im im) -0.5)
     (if (<= t_0 0.995)
       1.0
       (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 0.995) {
		tmp = 1.0;
	} else {
		tmp = fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 0.995)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(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[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.995], 1.0, N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 22.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 95.7%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto 1 \]

   if 0.994999999999999996 < (*.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 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 74.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 22.9%

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

   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 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 70.6%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.7%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

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

Alternative 15: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 16: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\cos im}{\mathsf{fma}\left(re \cdot re, -0.5, re - 1\right)}\\ \mathbf{if}\;re \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -500:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (- (cos im)) (fma (* re re) -0.5 (- re 1.0)))))
   (if (<= re -1.35e+154)
     t_0
     (if (<= re -500.0)
       (* (exp re) (* (* im im) -0.5))
       (if (<= re 3.5e-5)
         t_0
         (if (<= re 6.5e+86)
           (* (exp re) (fma (* im im) -0.5 1.0))
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (cos im))))))))

C

double code(double re, double im) {
	double t_0 = -cos(im) / fma((re * re), -0.5, (re - 1.0));
	double tmp;
	if (re <= -1.35e+154) {
		tmp = t_0;
	} else if (re <= -500.0) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (re <= 3.5e-5) {
		tmp = t_0;
	} else if (re <= 6.5e+86) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(-cos(im)) / fma(Float64(re * re), -0.5, Float64(re - 1.0)))
	tmp = 0.0
	if (re <= -1.35e+154)
		tmp = t_0;
	elseif (re <= -500.0)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (re <= 3.5e-5)
		tmp = t_0;
	elseif (re <= 6.5e+86)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-N[Cos[im], $MachinePrecision]) / N[(N[(re * re), $MachinePrecision] * -0.5 + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.35e+154], t$95$0, If[LessEqual[re, -500.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.5e-5], t$95$0, If[LessEqual[re, 6.5e+86], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.35000000000000003e154 or -500 < re < 3.4999999999999997e-5

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.35000000000000003e154 < re < -500

   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 84.6

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

   5. Applied rewrites 84.6%

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

   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 84.6%

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

   if 3.4999999999999997e-5 < re < 6.49999999999999996e86

   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 81.0

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

   5. Applied rewrites 81.0%

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

   if 6.49999999999999996e86 < 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 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 17: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -500.0)
   (* (exp re) (* (* im im) -0.5))
   (if (<= re 3.5e-5)
     (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
     (if (<= re 6.5e+86)
       (* (exp re) (fma (* im im) -0.5 1.0))
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (cos im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -500.0) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (re <= 3.5e-5) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 6.5e+86) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -500.0)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (re <= 3.5e-5)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 6.5e+86)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -500.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.5e-5], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+86], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -500

   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 73.1

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

   5. Applied rewrites 73.1%

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

   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 73.1%

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

   if -500 < re < 3.4999999999999997e-5

   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 3.4999999999999997e-5 < re < 6.49999999999999996e86

   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 81.0

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

   5. Applied rewrites 81.0%

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

   if 6.49999999999999996e86 < 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 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 18: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-5} \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -500.0)
   (* (exp re) (* (* im im) -0.5))
   (if (or (<= re 3.5e-5) (not (<= re 1.9e+154)))
     (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
     (* (exp re) (fma (* im im) -0.5 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -500.0) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if ((re <= 3.5e-5) || !(re <= 1.9e+154)) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -500.0)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((re <= 3.5e-5) || !(re <= 1.9e+154))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	else
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -500.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 3.5e-5], N[Not[LessEqual[re, 1.9e+154]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -500

   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 73.1

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

   5. Applied rewrites 73.1%

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

   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 73.1%

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

   if -500 < re < 3.4999999999999997e-5 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 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 3.4999999999999997e-5 < 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 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-5} \lor \neg \left(re \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 54.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -1.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{-t\_0}{\mathsf{fma}\left(re \cdot re, -0.5, re - 1\right)}\\ \mathbf{elif}\;re \leq -550:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)))
   (if (<= re -1.9e+185)
     (/ (- t_0) (fma (* re re) -0.5 (- re 1.0)))
     (if (<= re -550.0)
       (/ (- (* -0.5 (* im im))) (- re 1.0))
       (if (<= re 4.6e-11)
         1.0
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          t_0))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -1.9e+185) {
		tmp = -t_0 / fma((re * re), -0.5, (re - 1.0));
	} else if (re <= -550.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (re <= 4.6e-11) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	tmp = 0.0
	if (re <= -1.9e+185)
		tmp = Float64(Float64(-t_0) / fma(Float64(re * re), -0.5, Float64(re - 1.0)));
	elseif (re <= -550.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (re <= 4.6e-11)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * 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[re, -1.9e+185], N[((-t$95$0) / N[(N[(re * re), $MachinePrecision] * -0.5 + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -550.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e-11], 1.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.8999999999999999e185

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 6.8

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 6.8%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 4.0

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

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. associate--l+ N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 66.7

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

   13. Applied rewrites 66.7%

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

   if -1.8999999999999999e185 < re < -550

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 4.2

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 4.2%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.2

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

   10. Applied rewrites 3.2%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 34.6%

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

   if -550 < re < 4.60000000000000027e-11

   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.3%

       \[\leadsto 1 \]

   if 4.60000000000000027e-11 < 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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(re \cdot re, -0.5, re - 1\right)}\\ \mathbf{elif}\;re \leq -550:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -2.05 \cdot 10^{-14}:\\ \;\;\;\;\frac{-t\_0}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)))
   (if (<= re -2.05e-14)
     (/ (- t_0) (fma (* (- (* 0.16666666666666666 re) 0.5) re) re (- re 1.0)))
     (if (<= re 4.6e-11)
       1.0
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) t_0)))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -2.05e-14) {
		tmp = -t_0 / fma((((0.16666666666666666 * re) - 0.5) * re), re, (re - 1.0));
	} else if (re <= 4.6e-11) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	tmp = 0.0
	if (re <= -2.05e-14)
		tmp = Float64(Float64(-t_0) / fma(Float64(Float64(Float64(0.16666666666666666 * re) - 0.5) * re), re, Float64(re - 1.0)));
	elseif (re <= 4.6e-11)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * 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[re, -2.05e-14], N[((-t$95$0) / N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision] * re), $MachinePrecision] * re + N[(re - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e-11], 1.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.0500000000000001e-14

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.4

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

   10. Applied rewrites 49.4%

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

   if -2.0500000000000001e-14 < re < 4.60000000000000027e-11

   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.3%

       \[\leadsto 1 \]

   if 4.60000000000000027e-11 < 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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.05 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot re - 0.5\right) \cdot re, re, re - 1\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 51.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -550.0)
   (/ (- (* -0.5 (* im im))) (- re 1.0))
   (if (<= re 4.6e-11)
     1.0
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -550.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (re <= 4.6e-11) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -550.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (re <= 4.6e-11)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -550.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e-11], 1.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -550

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 5.1

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 5.1%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.5

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

   10. Applied rewrites 3.5%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 34.8%

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

   if -550 < re < 4.60000000000000027e-11

   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.3%

       \[\leadsto 1 \]

   if 4.60000000000000027e-11 < 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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 49.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -550.0)
   (/ (- (* -0.5 (* im im))) (- re 1.0))
   (if (<= re 4.6e-11)
     1.0
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -550.0) {
		tmp = -(-0.5 * (im * im)) / (re - 1.0);
	} else if (re <= 4.6e-11) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -550.0)
		tmp = Float64(Float64(-Float64(-0.5 * Float64(im * im))) / Float64(re - 1.0));
	elseif (re <= 4.6e-11)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -550.0], N[((-N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]) / N[(re - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e-11], 1.0, N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -550

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-\cos im}{\frac{-1}{e^{re}}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]
   6. Step-by-step derivation

      1. lower--.f64 5.1

         \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   7. Applied rewrites 5.1%

      \[\leadsto \frac{-\cos im}{\color{blue}{re - 1}} \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.5

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

   10. Applied rewrites 3.5%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 34.8%

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

   if -550 < re < 4.60000000000000027e-11

   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.3%

       \[\leadsto 1 \]

   if 4.60000000000000027e-11 < 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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 51.4

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

   8. Applied rewrites 51.4%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -550:\\ \;\;\;\;\frac{--0.5 \cdot \left(im \cdot im\right)}{re - 1}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 28.1% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. lower-cos.f64 56.3

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

5. Applied rewrites 56.3%

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

7. Applied rewrites 56.1%

   \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

8. Taylor expanded in im around 0

   \[\leadsto \cos \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 32.2%

    \[\leadsto 1 \]

11. Final simplification 32.2%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))