math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 21

Speedup: 1.0×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 87.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ t_2 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.99:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0))
        (t_2 (* (+ 1.0 re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       t_2
       (if (<= t_0 0.0)
         (* (exp re) (* (* im im) -0.5))
         (if (<= t_0 0.99)
           t_2
           (*
            t_1
            (fma
             (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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double t_2 = (1.0 + re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= 0.99) {
		tmp = t_2;
	} else {
		tmp = t_1 * fma(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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	t_2 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.99)
		tmp = t_2;
	else
		tmp = Float64(t_1 * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], t$95$2, If[LessEqual[t$95$0, 0.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99], t$95$2, N[(t$95$1 * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.98999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 0.98999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-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 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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 79.8%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 94.4

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

   11. Applied rewrites 94.4%

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

Alternative 3: 75.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ t_2 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.99:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0))
        (t_2 (* (+ 1.0 re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       t_2
       (if (<= t_0 0.0)
         (* 1.0 (* (* im im) -0.5))
         (if (<= t_0 0.99)
           t_2
           (*
            t_1
            (fma
             (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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double t_2 = (1.0 + re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else if (t_0 <= 0.99) {
		tmp = t_2;
	} else {
		tmp = t_1 * fma(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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	t_2 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.99)
		tmp = t_2;
	else
		tmp = Float64(t_1 * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], t$95$2, If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99], t$95$2, N[(t$95$1 * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.98999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.2%

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

   if 0.98999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-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 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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 79.8%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 94.4

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

   11. Applied rewrites 94.4%

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

Alternative 4: 75.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.99:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       (cos im)
       (if (<= t_0 0.0)
         (* 1.0 (* (* im im) -0.5))
         (if (<= t_0 0.99)
           (cos im)
           (*
            t_1
            (fma
             (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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else if (t_0 <= 0.99) {
		tmp = cos(im);
	} else {
		tmp = t_1 * fma(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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.99)
		tmp = cos(im);
	else
		tmp = Float64(t_1 * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99], N[Cos[im], $MachinePrecision], N[(t$95$1 * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.98999999999999999

   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%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.2%

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

   if 0.98999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-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 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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 79.8%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 94.4

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

   11. Applied rewrites 94.4%

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

Alternative 5: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.25}{\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
     (if (<= t_0 0.0)
       (* (exp re) (fma (* im im) -0.5 1.0))
       (if (<= t_0 0.99)
         (*
          (fma (fma (/ -0.25 (fma 0.16666666666666666 re -0.5)) re 1.0) re 1.0)
          (cos im))
         (*
          (exp re)
          (fma (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 <= -0.1) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.99) {
		tmp = fma(fma((-0.25 / fma(0.16666666666666666, re, -0.5)), re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re) * fma(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 <= -0.1)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif (t_0 <= 0.0)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.99)
		tmp = Float64(fma(fma(Float64(-0.25 / fma(0.16666666666666666, re, -0.5)), re, 1.0), re, 1.0) * cos(im));
	else
		tmp = Float64(exp(re) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99], N[(N[(N[(N[(-0.25 / N[(0.16666666666666666 * re + -0.5), $MachinePrecision]), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   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 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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%

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

   if -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.98999999999999999

   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 95.9

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{4}}{\mathsf{fma}\left(\frac{1}{6}, re, \frac{-1}{2}\right)}, re, 1\right), re, 1\right) \cdot \cos im \]
   9. Step-by-step derivation

   10. Applied rewrites 96.4%

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

   if 0.98999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 6: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
     (if (<= t_0 0.0)
       (* (exp re) (fma (* im im) -0.5 1.0))
       (if (<= t_0 0.99)
         (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
         (*
          (exp re)
          (fma (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 <= -0.1) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.99) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re) * fma(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 <= -0.1)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif (t_0 <= 0.0)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.99)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	else
		tmp = Float64(exp(re) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99], 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[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   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 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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%

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

   if -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.98999999999999999

   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 96.4

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

   5. Applied rewrites 96.4%

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

   if 0.98999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 7: 53.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.01)
       (* 1.0 (* (* im im) -0.5))
       (*
        t_1
        (fma (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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= 0.01) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = t_1 * fma(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 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.01)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = Float64(t_1 * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0100000000000000002

   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 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 15.5%

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

   if 0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-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 71.0

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

   5. Applied rewrites 71.0%

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 79.5

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

   11. Applied rewrites 79.5%

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

Alternative 8: 52.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      1.0
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.01)
       (* 1.0 (* (* im im) -0.5))
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= 0.01) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(1.0 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.01)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(1.0 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 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 re around 0

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

   8. Applied rewrites 48.7%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0100000000000000002

   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 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 15.5%

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

   if 0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-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 71.0

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

   5. Applied rewrites 71.0%

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 79.5

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

   11. Applied rewrites 79.5%

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

Alternative 9: 52.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      1.0
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.01)
       (* 1.0 (* (* im im) -0.5))
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (fma (* 0.041666666666666664 (* im im)) (* 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 = 1.0 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= 0.01) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((0.041666666666666664 * (im * im)), (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(1.0 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.01)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(0.041666666666666664 * Float64(im * im)), 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[(1.0 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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 re around 0

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

   8. Applied rewrites 48.7%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0100000000000000002

   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 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 15.5%

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

   if 0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-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 71.0

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

   5. Applied rewrites 71.0%

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 79.5

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

   11. Applied rewrites 79.5%

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

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 79.1%

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

Alternative 10: 40.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 18.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 57.1

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

   11. Applied rewrites 57.1%

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

Alternative 11: 46.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (* 1.0 (* (* im im) -0.5))
   (*
    (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 (exp(re) <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} 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 (exp(re) <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	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[N[Exp[re], $MachinePrecision], 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 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 2 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.2%

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

   if 0.0 < (exp.f64 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 61.7

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

   5. Applied rewrites 61.7%

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 57.6%

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

Alternative 12: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;re \leq 420000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.2)
   (* (exp re) (* (* im im) -0.5))
   (if (<= re 420000.0)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
     (if (<= re 1e+103)
       (* (exp re) (fma (* im im) -0.5 1.0))
       (* (* (* (fma 0.16666666666666666 re 0.5) re) re) (cos im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.2) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (re <= 420000.0) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 1e+103) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.2)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (re <= 420000.0)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.2], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 420000.0], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], 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), $MachinePrecision] * re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.20000000000000018

   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 -5.20000000000000018 < re < 4.2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 4.2e5 < re < 1e103

   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 76.9

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

   5. Applied rewrites 76.9%

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

   if 1e103 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 13: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -0.00011:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 420000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
   (if (<= re -0.00011)
     t_0
     (if (<= re 420000.0)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (<= re 1e+103)
         t_0
         (* (* (* (fma 0.16666666666666666 re 0.5) re) re) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -0.00011) {
		tmp = t_0;
	} else if (re <= 420000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 1e+103) {
		tmp = t_0;
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -0.00011)
		tmp = t_0;
	elseif (re <= 420000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 1e+103)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00011], t$95$0, If[LessEqual[re, 420000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], t$95$0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.10000000000000004e-4 or 4.2e5 < re < 1e103

   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 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.10000000000000004e-4 < re < 4.2e5

   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 98.5

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

   5. Applied rewrites 98.5%

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

   if 1e103 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 14: 46.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.2%

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

   if 0.0 < (exp.f64 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 61.7

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

   5. Applied rewrites 61.7%

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 57.0%

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

Alternative 15: 46.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.2%

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

   if 0.0 < (exp.f64 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 61.7

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

   5. Applied rewrites 61.7%

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

   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

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 57.6%

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

   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 57.0%

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

   13. Applied rewrites 57.0%

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

Alternative 16: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -0.00011:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 420000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
   (if (<= re -0.00011)
     t_0
     (if (<= re 420000.0)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (<= re 1.35e+154) t_0 (* (* (* re re) 0.5) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -0.00011) {
		tmp = t_0;
	} else if (re <= 420000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = ((re * re) * 0.5) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -0.00011)
		tmp = t_0;
	elseif (re <= 420000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00011], t$95$0, If[LessEqual[re, 420000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], t$95$0, N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.10000000000000004e-4 or 4.2e5 < re < 1.35000000000000003e154

   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 76.5

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

   5. Applied rewrites 76.5%

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

   if -1.10000000000000004e-4 < re < 4.2e5

   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 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.35000000000000003e154 < 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%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 17: 90.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -0.0001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 420000:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
   (if (<= re -0.0001)
     t_0
     (if (<= re 420000.0)
       (* (+ 1.0 re) (cos im))
       (if (<= re 1.35e+154) t_0 (* (* (* re re) 0.5) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -0.0001) {
		tmp = t_0;
	} else if (re <= 420000.0) {
		tmp = (1.0 + re) * cos(im);
	} else if (re <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = ((re * re) * 0.5) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -0.0001)
		tmp = t_0;
	elseif (re <= 420000.0)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	elseif (re <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0001], t$95$0, If[LessEqual[re, 420000.0], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], t$95$0, N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.00000000000000005e-4 or 4.2e5 < re < 1.35000000000000003e154

   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 76.5

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

   5. Applied rewrites 76.5%

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

   if -1.00000000000000005e-4 < re < 4.2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.35000000000000003e154 < 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%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 18: 44.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.01:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.01)
   (* 1.0 (* (* im im) -0.5))
   (* (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 (exp(re) <= 0.01) {
		tmp = 1.0 * ((im * im) * -0.5);
	} 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 (exp(re) <= 0.01)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	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[N[Exp[re], $MachinePrecision], 0.01], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 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 2 regimes

2. if (exp.f64 re) < 0.0100000000000000002

   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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 19.9%

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

   if 0.0100000000000000002 < (exp.f64 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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. rgt-mult-inverse N/A

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

      13. associate-*l* N/A

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

      14. unpow2 N/A

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

      15. cube-mult N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

   8. Applied rewrites 56.2%

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

Alternative 19: 37.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 re around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.2%

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

   if 0.0 < (exp.f64 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 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 43.9

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

   8. Applied rewrites 43.9%

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

Alternative 20: 35.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -11200000:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -11200000.0)
   (* 1.0 (* (* im im) -0.5))
   (* 1.0 (fma (* im im) -0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -11200000.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = 1.0 * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -11200000.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = Float64(1.0 * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -11200000.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.12e7

   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 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 20.7%

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

   if -1.12e7 < 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 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.6%

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

Alternative 21: 11.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 1.0 (* (* im im) -0.5)))

C

double code(double re, double im) {
	return 1.0 * ((im * im) * -0.5);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 * ((im * im) * (-0.5d0))
end function

Java

public static double code(double re, double im) {
	return 1.0 * ((im * im) * -0.5);
}

Python

def code(re, im):
	return 1.0 * ((im * im) * -0.5)

Julia

function code(re, im)
	return Float64(1.0 * Float64(Float64(im * im) * -0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 * ((im * im) * -0.5);
end

Wolfram

code[re_, im_] := N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

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 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.9%

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

9. Taylor expanded in im around inf

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

11. Applied rewrites 9.4%

    \[\leadsto 1 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{-0.5}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))