Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.5%

Time: 9.1s

Alternatives: 7

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (exp (- (fabs (- n m)) (+ (pow (fma 0.5 (+ n m) (- M)) 2.0) l))) (cos M)))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l))) * cos(M);
}

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))) * cos(M))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

5. Applied rewrites 96.0%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

6. Final simplification 96.0%

   \[\leadsto e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M \]
7. Add Preprocessing

Alternative 2: 73.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{if}\;M \leq -2.45 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq -2.9 \cdot 10^{-186}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (exp (* (- M) M)) (cos M))))
   (if (<= M -2.45e+36)
     t_0
     (if (<= M -2.9e-186)
       (* (cos M) (exp (- l)))
       (if (<= M 27.0)
         (* (exp (* (* n n) -0.25)) (fma (* M M) -0.5 1.0))
         t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-M * M)) * cos(M);
	double tmp;
	if (M <= -2.45e+36) {
		tmp = t_0;
	} else if (M <= -2.9e-186) {
		tmp = cos(M) * exp(-l);
	} else if (M <= 27.0) {
		tmp = exp(((n * n) * -0.25)) * fma((M * M), -0.5, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(-M) * M)) * cos(M))
	tmp = 0.0
	if (M <= -2.45e+36)
		tmp = t_0;
	elseif (M <= -2.9e-186)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (M <= 27.0)
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * fma(Float64(M * M), -0.5, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -2.45e+36], t$95$0, If[LessEqual[M, -2.9e-186], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 27.0], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.4499999999999999e36 or 27 < M

   1. Initial program 76.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in M around inf

      \[\leadsto e^{-1 \cdot {M}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

      \[\leadsto e^{\left(-M\right) \cdot M} \cdot \cos M \]

   if -2.4499999999999999e36 < M < -2.90000000000000019e-186

   1. Initial program 76.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 53.1

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

      2. lower-cos.f64 56.2

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

   8. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

   if -2.90000000000000019e-186 < M < 27

   1. Initial program 77.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 57.8%

      \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M \]

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 57.8%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 65.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8600:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \mathbf{elif}\;m \leq -1.36 \cdot 10^{-173}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -8600.0)
   (* (exp (* -0.25 (* m m))) (cos M))
   (if (<= m -1.36e-173)
     (* (exp (* (- M) M)) (cos M))
     (* (exp (* (* n n) -0.25)) (cos M)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8600.0) {
		tmp = exp((-0.25 * (m * m))) * cos(M);
	} else if (m <= -1.36e-173) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = exp(((n * n) * -0.25)) * cos(M);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-8600.0d0)) then
        tmp = exp(((-0.25d0) * (m * m))) * cos(m_1)
    else if (m <= (-1.36d-173)) then
        tmp = exp((-m_1 * m_1)) * cos(m_1)
    else
        tmp = exp(((n * n) * (-0.25d0))) * cos(m_1)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8600.0) {
		tmp = Math.exp((-0.25 * (m * m))) * Math.cos(M);
	} else if (m <= -1.36e-173) {
		tmp = Math.exp((-M * M)) * Math.cos(M);
	} else {
		tmp = Math.exp(((n * n) * -0.25)) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -8600.0:
		tmp = math.exp((-0.25 * (m * m))) * math.cos(M)
	elif m <= -1.36e-173:
		tmp = math.exp((-M * M)) * math.cos(M)
	else:
		tmp = math.exp(((n * n) * -0.25)) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -8600.0)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * cos(M));
	elseif (m <= -1.36e-173)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -8600.0)
		tmp = exp((-0.25 * (m * m))) * cos(M);
	elseif (m <= -1.36e-173)
		tmp = exp((-M * M)) * cos(M);
	else
		tmp = exp(((n * n) * -0.25)) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -8600.0], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.36e-173], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8600

   1. Initial program 69.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 96.5%

      \[\leadsto e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M \]

   if -8600 < m < -1.3600000000000001e-173

   1. Initial program 79.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 89.7%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in M around inf

      \[\leadsto e^{-1 \cdot {M}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 52.6%

      \[\leadsto e^{\left(-M\right) \cdot M} \cdot \cos M \]

   if -1.3600000000000001e-173 < m

   1. Initial program 78.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

      \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 60.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8600:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-186}:\\ \;\;\;\;e^{\left(-M\right) \cdot M} \cdot \cos M\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -8600.0)
   (* (exp (* -0.25 (* m m))) (cos M))
   (if (<= m 1.15e-186)
     (* (exp (* (- M) M)) (cos M))
     (* (exp (* (* n n) -0.25)) (fma (* M M) -0.5 1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8600.0) {
		tmp = exp((-0.25 * (m * m))) * cos(M);
	} else if (m <= 1.15e-186) {
		tmp = exp((-M * M)) * cos(M);
	} else {
		tmp = exp(((n * n) * -0.25)) * fma((M * M), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -8600.0)
		tmp = Float64(exp(Float64(-0.25 * Float64(m * m))) * cos(M));
	elseif (m <= 1.15e-186)
		tmp = Float64(exp(Float64(Float64(-M) * M)) * cos(M));
	else
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * fma(Float64(M * M), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -8600.0], N[(N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15e-186], N[(N[Exp[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8600

   1. Initial program 69.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in m around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {m}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 96.5%

      \[\leadsto e^{-0.25 \cdot \left(m \cdot m\right)} \cdot \cos M \]

   if -8600 < m < 1.15e-186

   1. Initial program 82.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in M around inf

      \[\leadsto e^{-1 \cdot {M}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 50.7%

      \[\leadsto e^{\left(-M\right) \cdot M} \cdot \cos M \]

   if 1.15e-186 < m

   1. Initial program 75.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 59.0%

      \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M \]

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 47.4%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+47} \lor \neg \left(n \leq 61\right):\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -1.55e+47) (not (<= n 61.0)))
   (* (exp (* (* n n) -0.25)) (fma (* M M) -0.5 1.0))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.55e+47) || !(n <= 61.0)) {
		tmp = exp(((n * n) * -0.25)) * fma((M * M), -0.5, 1.0);
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -1.55e+47) || !(n <= 61.0))
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * fma(Float64(M * M), -0.5, 1.0));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -1.55e+47], N[Not[LessEqual[n, 61.0]], $MachinePrecision]], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.55e47 or 61 < n

   1. Initial program 70.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

      \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M \]

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 74.0%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]

   if -1.55e47 < n < 61

   1. Initial program 82.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 43.4

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

      2. lower-cos.f64 47.6

         \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]

   8. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+47} \lor \neg \left(n \leq 61\right):\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+47} \lor \neg \left(n \leq 61\right):\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -1.55e+47) (not (<= n 61.0)))
   (* (exp (* (* n n) -0.25)) (fma (* M M) -0.5 1.0))
   (* 1.0 (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.55e+47) || !(n <= 61.0)) {
		tmp = exp(((n * n) * -0.25)) * fma((M * M), -0.5, 1.0);
	} else {
		tmp = 1.0 * exp(-l);
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -1.55e+47) || !(n <= 61.0))
		tmp = Float64(exp(Float64(Float64(n * n) * -0.25)) * fma(Float64(M * M), -0.5, 1.0));
	else
		tmp = Float64(1.0 * exp(Float64(-l)));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -1.55e+47], N[Not[LessEqual[n, 61.0]], $MachinePrecision]], N[(N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.55e47 or 61 < n

   1. Initial program 70.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right) \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}\right)} \cdot \cos \left(\mathsf{neg}\left(M\right)\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \cdot \cos M} \]

   6. Taylor expanded in n around inf

      \[\leadsto e^{\frac{-1}{4} \cdot {n}^{2}} \cdot \cos M \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

      \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \cos M \]

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 74.0%

       \[\leadsto e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, \color{blue}{-0.5}, 1\right) \]

   if -1.55e47 < n < 61

   1. Initial program 82.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 43.4

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

   6. Taylor expanded in M around 0

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right) + M \cdot \sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{-\ell} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(M \cdot \sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right) + \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{-\ell} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right) \cdot M} + \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{-\ell} \]

      4. lower-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}, M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}, M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}\right) \cdot e^{-\ell} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}\right) \cdot e^{-\ell} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right)\right) \cdot e^{-\ell} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right)\right) \cdot e^{-\ell} \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right)\right) \cdot e^{-\ell} \]

      15. lower-+.f64 42.5

          \[\leadsto \mathsf{fma}\left(\sin \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(0.5 \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right)\right) \cdot e^{-\ell} \]

   8. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)\right)} \cdot e^{-\ell} \]

   9. Taylor expanded in K around 0

      \[\leadsto 1 \cdot e^{-\ell} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+47} \lor \neg \left(n \leq 61\right):\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25} \cdot \mathsf{fma}\left(M \cdot M, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* 1.0 (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 * exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 * Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return 1.0 * math.exp(-l)

Julia

function code(K, m, n, M, l)
	return Float64(1.0 * exp(Float64(-l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 * exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in l around inf

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

   2. lower-neg.f64 33.0

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

5. Applied rewrites 33.0%

   \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{-\ell}} \]

6. Taylor expanded in M around 0

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right) + M \cdot \sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{-\ell} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(M \cdot \sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right) + \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{-\ell} \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right) \cdot M} + \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right)} \cdot e^{-\ell} \]

   4. lower-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}, M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}, M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   9. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \cdot e^{-\ell} \]

   10. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}\right) \cdot e^{-\ell} \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)}\right) \cdot e^{-\ell} \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right)\right) \cdot e^{-\ell} \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \color{blue}{\left(\left(m + n\right) \cdot K\right)}\right)\right) \cdot e^{-\ell} \]

   14. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(\frac{1}{2} \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right)\right) \cdot e^{-\ell} \]

   15. lower-+.f64 32.4

       \[\leadsto \mathsf{fma}\left(\sin \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(0.5 \cdot \left(\color{blue}{\left(n + m\right)} \cdot K\right)\right)\right) \cdot e^{-\ell} \]

8. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right), M, \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)\right)} \cdot e^{-\ell} \]

9. Taylor expanded in K around 0

   \[\leadsto 1 \cdot e^{-\ell} \]
10. Step-by-step derivation

11. Applied rewrites 40.1%

    \[\leadsto 1 \cdot e^{-\ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024303 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))