math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 17

Speedup: 1.5×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return (Math.exp(im) + Math.exp(-im)) * (Math.cos(re) * 0.5);
}

Python

def code(re, im):
	return (math.exp(im) + math.exp(-im)) * (math.cos(re) * 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \]
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot \cosh im\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999998:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (* (* re re) -0.5) (cosh im))
     (if (<= t_1 0.9999999999999998)
       (* (fma im im 2.0) t_0)
       (* 1.0 (cosh im))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((re * re) * -0.5) * cosh(im);
	} else if (t_1 <= 0.9999999999999998) {
		tmp = fma(im, im, 2.0) * t_0;
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * -0.5) * cosh(im));
	elseif (t_1 <= 0.9999999999999998)
		tmp = Float64(fma(im, im, 2.0) * t_0);
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999998], N[(N[(im * im + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in re around inf

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

   12. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999978

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 0.99999999999999978 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot \cosh im\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9999999999999998:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot \cosh im\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999998:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (* (* (* re re) -0.5) (cosh im))
     (if (<= t_0 0.9999999999999998) (cos re) (* 1.0 (cosh im))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * -0.5) * cosh(im);
	} else if (t_0 <= 0.9999999999999998) {
		tmp = cos(re);
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(im) + Math.exp(-im)) * (Math.cos(re) * 0.5);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((re * re) * -0.5) * Math.cosh(im);
	} else if (t_0 <= 0.9999999999999998) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.0 * Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(im) + math.exp(-im)) * (math.cos(re) * 0.5)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((re * re) * -0.5) * math.cosh(im)
	elif t_0 <= 0.9999999999999998:
		tmp = math.cos(re)
	else:
		tmp = 1.0 * math.cosh(im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * -0.5) * cosh(im));
	elseif (t_0 <= 0.9999999999999998)
		tmp = cos(re);
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((re * re) * -0.5) * cosh(im);
	elseif (t_0 <= 0.9999999999999998)
		tmp = cos(re);
	else
		tmp = 1.0 * cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999998], N[Cos[re], $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in re around inf

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

   12. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999978

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.99999999999999978 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot \cosh im\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9999999999999998:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999998:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma -0.5 (* re re) 1.0)
      (fma
       (fma
        (fma 0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.9999999999999998) (cos re) (* 1.0 (cosh im))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (re * re), 1.0) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	} else if (t_0 <= 0.9999999999999998) {
		tmp = cos(re);
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.9999999999999998)
		tmp = cos(re);
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999998], N[Cos[re], $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 96.7

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

   12. Applied rewrites 96.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999978

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.99999999999999978 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9999999999999998:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 -0.004)
     (fma -0.5 (* re re) 1.0)
     (if (<= t_0 2.0) 1.0 (* (* im im) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -0.004) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= -0.004)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(im * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.004], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.1%

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

   if -0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.0%

      \[\leadsto 1 \]

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 57.7

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 57.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 57.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.1:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) -0.1)
   (* (* -0.25 (* re re)) (* im im))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= -0.1) {
		tmp = (-0.25 * (re * re)) * (im * im);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= -0.1)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * Float64(im * im));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.6

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 75.6%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. lower-*.f64 45.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 44.8%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \color{blue}{im}\right) \]

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 44.8%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \left(im \cdot im\right) \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 79.9

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 79.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 65.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.1:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) 2.0)
   1.0
   (* (* im im) 0.5)))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((exp(im) + exp(-im)) * (cos(re) * 0.5d0)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (im * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.exp(im) + Math.exp(-im)) * (Math.cos(re) * 0.5)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.exp(im) + math.exp(-im)) * (math.cos(re) * 0.5)) <= 2.0:
		tmp = 1.0
	else:
		tmp = (im * im) * 0.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= 2.0)
		tmp = 1.0;
	else
		tmp = (im * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.7%

      \[\leadsto 1 \]

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 57.7

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 57.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 57.7%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{im} + e^{-im} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (+ (exp im) (exp (- im))) 5e+15)
   (cos re)
   (*
    (fma -0.5 (* re re) 1.0)
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(im) + exp(-im)) <= 5e+15) {
		tmp = cos(re);
	} else {
		tmp = fma(-0.5, (re * re), 1.0) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(im) + exp(Float64(-im))) <= 5e+15)
		tmp = cos(re);
	else
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision], 5e+15], N[Cos[re], $MachinePrecision], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5e15

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 5e15 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.8

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

   9. Applied rewrites 82.8%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 74.2

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

   12. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{im} + e^{-im} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (fma -0.5 (* re re) 1.0)
          (fma
           (fma
            (fma 0.001388888888888889 (* im im) 0.041666666666666664)
            (* im im)
            0.5)
           (* im im)
           1.0))))
   (if (<= (cos re) -0.004)
     t_0
     (if (<= (cos re) 0.999996) (* 0.5 (fma im im 2.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = fma(-0.5, (re * re), 1.0) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = t_0;
	} else if (cos(re) <= 0.999996) {
		tmp = 0.5 * fma(im, im, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(-0.5, Float64(re * re), 1.0) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0))
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = t_0;
	elseif (cos(re) <= 0.999996)
		tmp = Float64(0.5 * fma(im, im, 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.999996], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0040000000000000001 or 0.999995999999999996 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.1

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 80.4

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

   12. Applied rewrites 80.4%

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

   if -0.0040000000000000001 < (cos.f64 re) < 0.999995999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445 \cdot \left(re \cdot re\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.004)
   (*
    (fma
     (fma (* -0.0006944444444444445 (* re re)) (* re re) -0.25)
     (* re re)
     0.5)
    (fma im im 2.0))
   (if (<= (cos re) 0.999996)
     (* 0.5 (fma im im 2.0))
     (*
      (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0)
      (fma -0.5 (* re re) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = fma(fma((-0.0006944444444444445 * (re * re)), (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.999996) {
		tmp = 0.5 * fma(im, im, 2.0);
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = Float64(fma(fma(Float64(-0.0006944444444444445 * Float64(re * re)), Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.999996)
		tmp = Float64(0.5 * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.999996], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.9

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 76.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      15. lower-*.f64 47.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440} \cdot {re}^{2}, re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 47.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445 \cdot \left(re \cdot re\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0040000000000000001 < (cos.f64 re) < 0.999995999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.7%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 0.999995999999999996 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 87.5

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

   12. Applied rewrites 87.5%

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

Alternative 11: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.004)
   (*
    (fma
     (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
     (* re re)
     0.5)
    (fma im im 2.0))
   (if (<= (cos re) 0.999996)
     (* 0.5 (fma im im 2.0))
     (*
      (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0)
      (fma -0.5 (* re re) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re), (re * re), 0.5) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.999996) {
		tmp = 0.5 * fma(im, im, 2.0);
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re), Float64(re * re), 0.5) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.999996)
		tmp = Float64(0.5 * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.999996], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.9

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 76.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      15. lower-*.f64 47.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{4} \cdot \left(\frac{1}{48} \cdot \frac{1}{{re}^{2}} - \frac{1}{1440}\right), \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 47.6%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re, \color{blue}{re} \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0040000000000000001 < (cos.f64 re) < 0.999995999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.7%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 0.999995999999999996 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 87.5

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

   12. Applied rewrites 87.5%

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

Alternative 12: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\left(im \cdot im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445 \cdot \left(re \cdot re\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.004)
   (*
    (* im im)
    (fma
     (fma (* -0.0006944444444444445 (* re re)) (* re re) -0.25)
     (* re re)
     0.5))
   (if (<= (cos re) 0.999996)
     (* 0.5 (fma im im 2.0))
     (*
      (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0)
      (fma -0.5 (* re re) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = (im * im) * fma(fma((-0.0006944444444444445 * (re * re)), (re * re), -0.25), (re * re), 0.5);
	} else if (cos(re) <= 0.999996) {
		tmp = 0.5 * fma(im, im, 2.0);
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = Float64(Float64(im * im) * fma(fma(Float64(-0.0006944444444444445 * Float64(re * re)), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (cos(re) <= 0.999996)
		tmp = Float64(0.5 * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], N[(N[(im * im), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.999996], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.9

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 76.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      15. lower-*.f64 47.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \left(im \cdot \color{blue}{im}\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440} \cdot {re}^{2}, re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot \left(im \cdot im\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 47.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445 \cdot \left(re \cdot re\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \left(im \cdot im\right) \]

   if -0.0040000000000000001 < (cos.f64 re) < 0.999995999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.7%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 0.999995999999999996 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 87.5

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

   12. Applied rewrites 87.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\left(im \cdot im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445 \cdot \left(re \cdot re\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.999996:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0)
          (fma -0.5 (* re re) 1.0))))
   (if (<= (cos re) -0.004)
     t_0
     (if (<= (cos re) 0.999996) (* 0.5 (fma im im 2.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = t_0;
	} else if (cos(re) <= 0.999996) {
		tmp = 0.5 * fma(im, im, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0))
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = t_0;
	elseif (cos(re) <= 0.999996)
		tmp = Float64(0.5 * fma(im, im, 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.999996], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0040000000000000001 or 0.999995999999999996 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.1

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 75.8

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

   12. Applied rewrites 75.8%

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

   if -0.0040000000000000001 < (cos.f64 re) < 0.999995999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.0

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.7%

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

Alternative 14: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. cosh-undef N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]
7. Add Preprocessing

Alternative 15: 58.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.004)
   (* (* -0.25 (* re re)) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = (-0.25 * (re * re)) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.9

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 76.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. lower-*.f64 43.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 43.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0040000000000000001 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 79.6

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 79.6%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.9%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.004) (fma -0.5 (* re re) 1.0) (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.004) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.004)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.004], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.1%

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

   if -0.0040000000000000001 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 79.6

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 79.6%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.9%

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

Alternative 17: 28.3% accurate, 316.0× 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%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. lower-cos.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 31.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))