math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 17.9s

Alternatives: 25

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* -0.5 (* im im)))
     (if (<= t_0 -0.05)
       (* (cos im) (+ re (fma (fma re 0.16666666666666666 0.5) (* re re) 1.0)))
       (if (<= t_0 1e-24)
         (exp re)
         (if (<= t_0 0.999999999975)
           (* (cos im) (fma re (fma re 0.5 1.0) 1.0))
           (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (-0.5 * (im * im));
	} else if (t_0 <= -0.05) {
		tmp = cos(im) * (re + fma(fma(re, 0.16666666666666666, 0.5), (re * re), 1.0));
	} else if (t_0 <= 1e-24) {
		tmp = exp(re);
	} else if (t_0 <= 0.999999999975) {
		tmp = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(-0.5 * Float64(im * im)));
	elseif (t_0 <= -0.05)
		tmp = Float64(cos(im) * Float64(re + fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), 1.0)));
	elseif (t_0 <= 1e-24)
		tmp = exp(re);
	elseif (t_0 <= 0.999999999975)
		tmp = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[Cos[im], $MachinePrecision] * N[(re + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-24], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.999999999975], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $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

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 96.7

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

   7. Applied rewrites 96.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 96.7

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

   9. Applied rewrites 96.7%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25 or 0.999999999975 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999975

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, 1\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* -0.5 (* im im)))
     (if (<= t_0 -0.05)
       (* (cos im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
       (if (<= t_0 1e-24)
         (exp re)
         (if (<= t_0 0.999999999975)
           (* (cos im) (fma re (fma re 0.5 1.0) 1.0))
           (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (-0.5 * (im * im));
	} else if (t_0 <= -0.05) {
		tmp = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 1e-24) {
		tmp = exp(re);
	} else if (t_0 <= 0.999999999975) {
		tmp = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(-0.5 * Float64(im * im)));
	elseif (t_0 <= -0.05)
		tmp = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 1e-24)
		tmp = exp(re);
	elseif (t_0 <= 0.999999999975)
		tmp = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-24], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.999999999975], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $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

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25 or 0.999999999975 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999975

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999999999975:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (fma re (fma re 0.5 1.0) 1.0)))
        (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* -0.5 (* im im)))
     (if (<= t_1 -0.05)
       t_0
       (if (<= t_1 1e-24)
         (exp re)
         (if (<= t_1 0.999999999975) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (-0.5 * (im * im));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 1e-24) {
		tmp = exp(re);
	} else if (t_1 <= 0.999999999975) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(-0.5 * Float64(im * im)));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 1e-24)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999975)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 1e-24], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999999999975], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999975

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25 or 0.999999999975 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re + 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.999999999975:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ re 1.0))) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* -0.5 (* im im)))
     (if (<= t_1 -0.05)
       t_0
       (if (<= t_1 1e-24)
         (exp re)
         (if (<= t_1 0.999999999975) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (re + 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (-0.5 * (im * im));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 1e-24) {
		tmp = exp(re);
	} else if (t_1 <= 0.999999999975) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (re + 1.0);
	double t_1 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (-0.5 * (im * im));
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 1e-24) {
		tmp = Math.exp(re);
	} else if (t_1 <= 0.999999999975) {
		tmp = t_0;
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (re + 1.0)
	t_1 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * (-0.5 * (im * im))
	elif t_1 <= -0.05:
		tmp = t_0
	elif t_1 <= 1e-24:
		tmp = math.exp(re)
	elif t_1 <= 0.999999999975:
		tmp = t_0
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(re + 1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(-0.5 * Float64(im * im)));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 1e-24)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999975)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (re + 1.0);
	t_1 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * (-0.5 * (im * im));
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 1e-24)
		tmp = exp(re);
	elseif (t_1 <= 0.999999999975)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 1e-24], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.999999999975], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999975

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25 or 0.999999999975 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.999999999975:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (cos im) (+ re 1.0))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (* im im)
       (fma
        (* im im)
        (fma (* im im) -0.001388888888888889 0.041666666666666664)
        -0.5)
       1.0))
     (if (<= t_0 -0.05)
       t_1
       (if (<= t_0 1e-24)
         (exp re)
         (if (<= t_0 0.999999999975) t_1 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * (re + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= -0.05) {
		tmp = t_1;
	} else if (t_0 <= 1e-24) {
		tmp = exp(re);
	} else if (t_0 <= 0.999999999975) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	elseif (t_0 <= -0.05)
		tmp = t_1;
	elseif (t_0 <= 1e-24)
		tmp = exp(re);
	elseif (t_0 <= 0.999999999975)
		tmp = t_1;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], t$95$1, If[LessEqual[t$95$0, 1e-24], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.999999999975], t$95$1, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.9

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

   8. Applied rewrites 91.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999975

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25 or 0.999999999975 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999999999975:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-24}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (* im im)
       (fma
        (* im im)
        (fma (* im im) -0.001388888888888889 0.041666666666666664)
        -0.5)
       1.0))
     (if (<= t_0 -0.05)
       (cos im)
       (if (<= t_0 1e-24) (exp re) (if (<= t_0 0.995) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if (t_0 <= 1e-24) {
		tmp = exp(re);
	} else if (t_0 <= 0.995) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif (t_0 <= 1e-24)
		tmp = exp(re);
	elseif (t_0 <= 0.995)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-24], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.995], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.9

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

   8. Applied rewrites 91.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25 or 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 8: 56.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \frac{\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right), -1\right)}{\mathsf{fma}\left(re, 0.5, -1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (* im im)
       (fma
        (* im im)
        (fma (* im im) -0.001388888888888889 0.041666666666666664)
        -0.5)
       1.0))
     (if (<= t_0 0.0)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (fma
        re
        (/
         (fma
          (fma re 0.16666666666666666 0.5)
          (* re (* re (fma re 0.16666666666666666 0.5)))
          -1.0)
         (fma re 0.5 -1.0))
        1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = fma(re, (fma(fma(re, 0.16666666666666666, 0.5), (re * (re * fma(re, 0.16666666666666666, 0.5))), -1.0) / fma(re, 0.5, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = fma(re, Float64(fma(fma(re, 0.16666666666666666, 0.5), Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5))), -1.0) / fma(re, 0.5, -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(re * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.6

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

   8. Applied rewrites 48.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 31.4

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

   11. Applied rewrites 31.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.9

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

   8. Applied rewrites 67.9%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 56.1

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

   10. Applied rewrites 56.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 70.2%

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

Alternative 9: 55.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.05)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma
       (* im im)
       (fma
        (* im im)
        (fma (* im im) -0.001388888888888889 0.041666666666666664)
        -0.5)
       1.0))
     (if (<= t_0 0.0)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma((im * im), fma((im * im), fma((im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.6

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

   8. Applied rewrites 48.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 31.4

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

   11. Applied rewrites 31.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.9

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

   8. Applied rewrites 67.9%

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

Alternative 10: 55.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 46.8

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

   8. Applied rewrites 46.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 31.4

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

   11. Applied rewrites 31.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.9

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

   8. Applied rewrites 67.9%

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

Alternative 11: 55.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 46.8

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. associate-*l* N/A

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

       11. lft-mult-inverse N/A

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

       12. metadata-eval N/A

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

       13. rgt-mult-inverse N/A

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

       14. distribute-lft-in N/A

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

   11. Applied rewrites 46.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 31.4

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

   11. Applied rewrites 31.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.9

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

   8. Applied rewrites 67.9%

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

Alternative 12: 55.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.06:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.06)
     (*
      (fma im (* im -0.5) 1.0)
      (* (fma re 0.16666666666666666 0.5) (* re re)))
     (if (<= t_0 0.0)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.06) {
		tmp = fma(im, (im * -0.5), 1.0) * (fma(re, 0.16666666666666666, 0.5) * (re * re));
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.06)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.06], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.059999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 47.8

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

   8. Applied rewrites 47.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 47.6

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

   11. Applied rewrites 47.6%

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

   if -0.059999999999999998 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 31.0

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

   11. Applied rewrites 31.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.9

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

   8. Applied rewrites 67.9%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.06:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.06:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.06)
     (*
      (* re (* re re))
      (fma (* im im) -0.08333333333333333 0.16666666666666666))
     (if (<= t_0 0.0)
       (*
        (* (* im im) (* im im))
        (fma re 0.041666666666666664 0.041666666666666664))
       (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.06) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, 0.16666666666666666);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.06)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.06], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.059999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 47.8

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

   8. Applied rewrites 47.8%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{1}{6}\right) \]

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

       17. lower-fma.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. metadata-eval 46.9

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right) \]

   11. Applied rewrites 46.9%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)} \]

   if -0.059999999999999998 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 31.0

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

   11. Applied rewrites 31.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.9

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

   8. Applied rewrites 67.9%

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

Alternative 14: 44.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 9.6

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

   8. Applied rewrites 9.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 70.5

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

   8. Applied rewrites 70.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 62.1

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

   8. Applied rewrites 62.1%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

          \[\leadsto re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       8. +-commutative N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       10. associate-*l* N/A

           \[\leadsto re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       11. lft-mult-inverse N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       12. metadata-eval N/A

           \[\leadsto re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       13. rgt-mult-inverse N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + \color{blue}{1}\right) \]

       14. +-commutative N/A

           \[\leadsto re \cdot \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]

   11. Applied rewrites 62.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)} \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \frac{1}{6}} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \frac{1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \frac{1}{6} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{re \cdot \left({re}^{2} \cdot \frac{1}{6}\right)} \]

       5. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right)} \]

       8. unpow2 N/A

          \[\leadsto re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       9. lower-*.f64 62.1

          \[\leadsto re \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   14. Applied rewrites 62.1%

       \[\leadsto \color{blue}{re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 41.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (fma -0.5 (* im im) 1.0)
     (if (<= t_0 2.0) (+ re 1.0) (fma (* re re) 0.5 re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = fma((re * re), 0.5, re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = fma(Float64(re * re), 0.5, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re + 1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5 + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 20.0

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 70.7

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. lower-+.f64 70.3

         \[\leadsto \color{blue}{1 + re} \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{1 + re} \]

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 100.0

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 62.1

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \]

       4. associate-+r+ N/A

          \[\leadsto re \cdot \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \frac{1}{{re}^{2}}\right)}\right) \]

       5. distribute-lft-in N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right)} \]

       6. unpow2 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       7. associate-*l* N/A

          \[\leadsto re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       8. +-commutative N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       10. associate-*l* N/A

           \[\leadsto re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       11. lft-mult-inverse N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       12. metadata-eval N/A

           \[\leadsto re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       13. rgt-mult-inverse N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + \color{blue}{1}\right) \]

       14. +-commutative N/A

           \[\leadsto re \cdot \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]

   11. Applied rewrites 62.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{2}}, re\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 50.2%

       \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{0.5}, re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.96:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.96)
   (*
    (* re (* re re))
    (fma (* im im) -0.08333333333333333 0.16666666666666666))
   (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.96) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, 0.16666666666666666);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.96)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.96], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.95999999999999996

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{re} + \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot e^{re}} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot e^{re}} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot e^{re} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{e^{re} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      7. unpow2 N/A

         \[\leadsto e^{re} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto e^{re} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot im\right) \cdot im} + 1\right) \]

      9. *-commutative N/A

         \[\leadsto e^{re} \cdot \left(\color{blue}{im \cdot \left(\frac{-1}{2} \cdot im\right)} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto e^{re} \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{2} \cdot im, 1\right)} \]

      11. *-commutative N/A

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{2}}, 1\right) \]

      12. lower-*.f64 92.5

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.5}, 1\right) \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      7. lower-fma.f64 81.3

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   8. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right) \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{3} \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]

       5. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot {re}^{2}\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)}\right) \]

       11. distribute-rgt-in N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{1}{6}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{1}{6}\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{-1}{2}\right)} + 1 \cdot \frac{1}{6}\right) \]

       16. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{2}\right) + \color{blue}{\frac{1}{6}}\right) \]

       17. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{6} \cdot \frac{-1}{2}, \frac{1}{6}\right)} \]

       18. unpow2 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{6} \cdot \frac{-1}{2}, \frac{1}{6}\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{6} \cdot \frac{-1}{2}, \frac{1}{6}\right) \]

       20. metadata-eval 81.5

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right) \]

   11. Applied rewrites 81.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)} \]

   if -0.95999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.0

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 41.3

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 10^{-24}:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 1e-24)
   (* (+ re 1.0) (fma -0.5 (* im im) 1.0))
   (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 1e-24) {
		tmp = (re + 1.0) * fma(-0.5, (im * im), 1.0);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 1e-24)
		tmp = Float64(Float64(re + 1.0) * fma(-0.5, Float64(im * im), 1.0));
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 1e-24], N[(N[(re + 1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999924e-25

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

      2. lower-+.f64 20.0

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 14.2

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 14.2%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 9.99999999999999924e-25 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.6

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.6%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 68.4

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma -0.5 (* im im) 1.0)
   (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	else
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 20.0

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.2

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 67.9

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma -0.5 (* im im) 1.0)
   (fma re (* re (fma re 0.16666666666666666 0.5)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else {
		tmp = fma(re, (re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	else
		tmp = fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 20.0

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.2

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 67.9

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right), 1\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, 1\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right), 1\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)}, 1\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right), 1\right) \]

       6. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right), 1\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right), 1\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right), 1\right) \]

       11. lower-fma.f64 67.4

           \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right) \]

   11. Applied rewrites 67.4%

       \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma -0.5 (* im im) 1.0)
   (fma re (* re (* re 0.16666666666666666)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else {
		tmp = fma(re, (re * (re * 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	else
		tmp = fma(re, Float64(re * Float64(re * 0.16666666666666666)), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 20.0

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.2

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 67.9

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot {re}^{2}}, 1\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \frac{1}{6} \cdot \color{blue}{\left(re \cdot re\right)}, 1\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(\frac{1}{6} \cdot re\right) \cdot re}, 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{6} \cdot re\right)}, 1\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{6} \cdot re\right)}, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \frac{1}{6}\right)}, 1\right) \]

       6. lower-*.f64 67.3

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}, 1\right) \]

   11. Applied rewrites 67.3%

       \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(re \cdot 0.16666666666666666\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 41.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma -0.5 (* im im) 1.0)
   (fma re (fma re 0.5 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	else
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 20.0

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.2

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + \frac{1}{2} \cdot re, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot re + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{2}} + 1, 1\right) \]

      5. lower-fma.f64 64.6

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.5, 1\right)}, 1\right) \]

   8. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 32.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0) (fma -0.5 (* im im) 1.0) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 20.0

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {im}^{2} + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. lower-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 79.2

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. lower-+.f64 51.4

         \[\leadsto \color{blue}{1 + re} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{1 + re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;re \leq -82000:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{+30}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, re \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}{\mathsf{fma}\left(re, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (fma re 0.16666666666666666 0.5) 1.0)))
   (if (<= re -82000.0)
     (*
      (* (* im im) (* im im))
      (fma re 0.041666666666666664 0.041666666666666664))
     (if (<= re 6.6e+30)
       (cos im)
       (if (<= re 6.4e+95)
         (/
          (fma
           t_0
           (* re (fma (fma re 0.16666666666666666 0.5) (* re re) re))
           -1.0)
          (fma re t_0 -1.0))
         (*
          (* re (* 0.16666666666666666 (* re re)))
          (fma im (* im -0.5) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (re <= -82000.0) {
		tmp = ((im * im) * (im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664);
	} else if (re <= 6.6e+30) {
		tmp = cos(im);
	} else if (re <= 6.4e+95) {
		tmp = fma(t_0, (re * fma(fma(re, 0.16666666666666666, 0.5), (re * re), re)), -1.0) / fma(re, t_0, -1.0);
	} else {
		tmp = (re * (0.16666666666666666 * (re * re))) * fma(im, (im * -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (re <= -82000.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * fma(re, 0.041666666666666664, 0.041666666666666664));
	elseif (re <= 6.6e+30)
		tmp = cos(im);
	elseif (re <= 6.4e+95)
		tmp = Float64(fma(t_0, Float64(re * fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re)), -1.0) / fma(re, t_0, -1.0));
	else
		tmp = Float64(Float64(re * Float64(0.16666666666666666 * Float64(re * re))) * fma(im, Float64(im * -0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[re, -82000.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.6e+30], N[Cos[im], $MachinePrecision], If[LessEqual[re, 6.4e+95], N[(N[(t$95$0 * N[(re * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(re * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -82000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

      2. lower-+.f64 2.2

         \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   5. Applied rewrites 2.2%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 1.9

          \[\leadsto \left(re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 1.9%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + re\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + re\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(re + 1\right)}\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot \frac{1}{24} + 1 \cdot \frac{1}{24}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \frac{1}{24} + \color{blue}{\frac{1}{24}}\right) \]

       15. lower-fma.f64 31.9

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   11. Applied rewrites 31.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664, 0.041666666666666664\right)} \]

   if -82000 < re < 6.60000000000000053e30

   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 90.8

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\cos im} \]

   if 6.60000000000000053e30 < re < 6.4000000000000001e95

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lower-exp.f64 92.3

         \[\leadsto \color{blue}{e^{re}} \]

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. lower-fma.f64 4.7

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto re \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)} + 1\right) + 1 \]

      2. lift-fma.f64 N/A

         \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)} + 1 \]

      3. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)\right) - 1 \cdot 1}{re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right) - 1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)\right) - 1 \cdot 1}{re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right) - 1}} \]

   10. Applied rewrites 55.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), re \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}} \]

   if 6.4000000000000001e95 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{re} + \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot e^{re}} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot e^{re}} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot e^{re} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{e^{re} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      7. unpow2 N/A

         \[\leadsto e^{re} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \]

      8. associate-*r* N/A

         \[\leadsto e^{re} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot im\right) \cdot im} + 1\right) \]

      9. *-commutative N/A

         \[\leadsto e^{re} \cdot \left(\color{blue}{im \cdot \left(\frac{-1}{2} \cdot im\right)} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto e^{re} \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{2} \cdot im, 1\right)} \]

      11. *-commutative N/A

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{2}}, 1\right) \]

      12. lower-*.f64 90.7

          \[\leadsto e^{re} \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.5}, 1\right) \]

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

      7. lower-fma.f64 88.6

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   8. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{3} \cdot \frac{1}{6}\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       2. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \frac{1}{6}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \frac{1}{6}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(re \cdot \left({re}^{2} \cdot \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       9. lower-*.f64 88.6

          \[\leadsto \left(re \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   11. Applied rewrites 88.6%

       \[\leadsto \color{blue}{\left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 24: 28.8% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. lower-exp.f64 71.3

      \[\leadsto \color{blue}{e^{re}} \]

5. Applied rewrites 71.3%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. lower-+.f64 28.5

      \[\leadsto \color{blue}{1 + re} \]

8. Applied rewrites 28.5%

   \[\leadsto \color{blue}{1 + re} \]

9. Final simplification 28.5%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 25: 28.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. lower-exp.f64 71.3

      \[\leadsto \color{blue}{e^{re}} \]

5. Applied rewrites 71.3%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 28.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))