Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.9%

Time: 20.2s

Alternatives: 9

Speedup: 1.3×

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: 76.1% 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: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(m_1) * exp(((abs((n - m)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-{M}^{2}}\\ \mathbf{if}\;m \leq -55:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-173}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (pow M 2.0))))))
   (if (<= m -55.0)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (if (<= m -2.6e-46)
       t_0
       (if (<= m -3.2e-56)
         (/ (cos M) (exp l))
         (if (<= m 3.1e-173) t_0 (* (cos M) (exp (* (pow n 2.0) -0.25)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-pow(M, 2.0));
	double tmp;
	if (m <= -55.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= -2.6e-46) {
		tmp = t_0;
	} else if (m <= -3.2e-56) {
		tmp = cos(M) / exp(l);
	} else if (m <= 3.1e-173) {
		tmp = t_0;
	} else {
		tmp = cos(M) * exp((pow(n, 2.0) * -0.25));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    if (m <= (-55.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= (-2.6d-46)) then
        tmp = t_0
    else if (m <= (-3.2d-56)) then
        tmp = cos(m_1) / exp(l)
    else if (m <= 3.1d-173) then
        tmp = t_0
    else
        tmp = cos(m_1) * exp(((n ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (m <= -55.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= -2.6e-46) {
		tmp = t_0;
	} else if (m <= -3.2e-56) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (m <= 3.1e-173) {
		tmp = t_0;
	} else {
		tmp = Math.cos(M) * Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	tmp = 0
	if m <= -55.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= -2.6e-46:
		tmp = t_0
	elif m <= -3.2e-56:
		tmp = math.cos(M) / math.exp(l)
	elif m <= 3.1e-173:
		tmp = t_0
	else:
		tmp = math.cos(M) * math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	tmp = 0.0
	if (m <= -55.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= -2.6e-46)
		tmp = t_0;
	elseif (m <= -3.2e-56)
		tmp = Float64(cos(M) / exp(l));
	elseif (m <= 3.1e-173)
		tmp = t_0;
	else
		tmp = Float64(cos(M) * exp(Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-(M ^ 2.0));
	tmp = 0.0;
	if (m <= -55.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= -2.6e-46)
		tmp = t_0;
	elseif (m <= -3.2e-56)
		tmp = cos(M) / exp(l);
	elseif (m <= 3.1e-173)
		tmp = t_0;
	else
		tmp = cos(M) * exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -55.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.6e-46], t$95$0, If[LessEqual[m, -3.2e-56], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.1e-173], t$95$0, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -55

   1. Initial program 72.9%

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

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in m around inf 96.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative 96.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   8. Simplified 96.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   if -55 < m < -2.6000000000000002e-46 or -3.19999999999999986e-56 < m < 3.10000000000000005e-173

   1. Initial program 81.7%

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

   3. Taylor expanded in K around 0 95.6%

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

      1. cos-neg 95.6%

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

   5. Simplified 95.6%

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

   6. Taylor expanded in M around inf 57.7%

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

      1. mul-1-neg 57.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   8. Simplified 57.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   if -2.6000000000000002e-46 < m < -3.19999999999999986e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf 35.2%

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

      1. mul-1-neg 35.2%

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

   5. Simplified 35.2%

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

   6. Taylor expanded in K around 0 35.2%

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

      1. exp-neg 35.2%

         \[\leadsto \cos \left(-M\right) \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

      2. associate-*r/ 35.2%

         \[\leadsto \color{blue}{\frac{\cos \left(-M\right) \cdot 1}{e^{\ell}}} \]

      3. *-rgt-identity 35.2%

         \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]

      4. cos-neg 35.2%

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

   8. Simplified 35.2%

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

   if 3.10000000000000005e-173 < m

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0 96.4%

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

      1. cos-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in n around inf 56.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative 56.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   8. Simplified 56.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;m \leq -3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-173}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5e+31)
   (*
    (cos M)
    (exp (+ (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) (- (fabs (- n m)) l))))
   (* (cos M) (exp (* (pow n 2.0) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5e+31) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (fabs((n - m)) - l)));
	} else {
		tmp = cos(M) * exp((pow(n, 2.0) * -0.25));
	}
	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 (n <= 5d+31) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) + (abs((n - m)) - l)))
    else
        tmp = cos(m_1) * exp(((n ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5e+31) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (Math.abs((n - m)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 5e+31:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (math.fabs((n - m)) - l)))
	else:
		tmp = math.cos(M) * math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5e+31)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) + Float64(abs(Float64(n - m)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 5e+31)
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) + (abs((n - m)) - l)));
	else
		tmp = cos(M) * exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5e+31], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.00000000000000027e31

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in n around 0 78.4%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 78.4%

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

      2. unpow2 78.4%

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

      3. distribute-rgt-out 82.0%

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

      4. *-commutative 82.0%

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

      5. *-commutative 82.0%

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

   8. Simplified 82.0%

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

   if 5.00000000000000027e31 < n

   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 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 98.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   8. Simplified 98.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.15 \cdot 10^{+25}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-173}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3.15e+25)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m 5e-173)
     (* (cos M) (exp (+ (* M (- n M)) (- (fabs (- n m)) l))))
     (* (cos M) (exp (* (pow n 2.0) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3.15e+25) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 5e-173) {
		tmp = cos(M) * exp(((M * (n - M)) + (fabs((n - m)) - l)));
	} else {
		tmp = cos(M) * exp((pow(n, 2.0) * -0.25));
	}
	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 <= (-3.15d+25)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 5d-173) then
        tmp = cos(m_1) * exp(((m_1 * (n - m_1)) + (abs((n - m)) - l)))
    else
        tmp = cos(m_1) * exp(((n ** 2.0d0) * (-0.25d0)))
    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 <= -3.15e+25) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 5e-173) {
		tmp = Math.cos(M) * Math.exp(((M * (n - M)) + (Math.abs((n - m)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3.15e+25:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 5e-173:
		tmp = math.cos(M) * math.exp(((M * (n - M)) + (math.fabs((n - m)) - l)))
	else:
		tmp = math.cos(M) * math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3.15e+25)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= 5e-173)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) + Float64(abs(Float64(n - m)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64((n ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3.15e+25)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 5e-173)
		tmp = cos(M) * exp(((M * (n - M)) + (abs((n - m)) - l)));
	else
		tmp = cos(M) * exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3.15e+25], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e-173], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.14999999999999987e25

   1. Initial program 72.7%

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

   3. Taylor expanded in K around 0 98.2%

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

      1. cos-neg 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in m around inf 96.4%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative 96.4%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   8. Simplified 96.4%

      \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   if -3.14999999999999987e25 < m < 5.0000000000000002e-173

   1. Initial program 82.6%

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

   3. Taylor expanded in K around 0 96.1%

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

      1. cos-neg 96.1%

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

   5. Simplified 96.1%

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

   6. Taylor expanded in n around 0 67.9%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 67.9%

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

      2. unpow2 67.9%

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

      3. distribute-rgt-out 70.1%

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

      4. *-commutative 70.1%

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

      5. *-commutative 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in m around 0 69.6%

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

       1. mul-1-neg 69.6%

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

       2. *-commutative 69.6%

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

       3. distribute-rgt-neg-in 69.6%

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

   11. Simplified 69.6%

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

   if 5.0000000000000002e-173 < m

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0 96.4%

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

      1. cos-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in n around inf 56.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative 56.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   8. Simplified 56.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.15 \cdot 10^{+25}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-173}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -26 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -26.0) (not (<= M 27.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (* -0.25 (pow m 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -26.0) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	}
	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_1 <= (-26.0d0)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    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 <= -26.0) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -26.0) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -26.0) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -26.0) || ~((M <= 27.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -26.0], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -26 or 27 < M

   1. Initial program 80.5%

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

   3. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in M around inf 95.6%

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

      1. mul-1-neg 95.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   8. Simplified 95.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   if -26 < M < 27

   1. Initial program 75.4%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in m around inf 51.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   7. Step-by-step derivation

      1. *-commutative 51.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

   8. Simplified 51.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -26 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -26 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -26.0) (not (<= M 27.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -26.0) || !(M <= 27.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) / exp(l);
	}
	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_1 <= (-26.0d0)) .or. (.not. (m_1 <= 27.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) / exp(l)
    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 <= -26.0) || !(M <= 27.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -26.0) or not (M <= 27.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -26.0) || !(M <= 27.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -26.0) || ~((M <= 27.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -26.0], N[Not[LessEqual[M, 27.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -26 or 27 < M

   1. Initial program 80.5%

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

   3. Taylor expanded in K around 0 98.5%

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

      1. cos-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in M around inf 95.6%

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

      1. mul-1-neg 95.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   8. Simplified 95.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   if -26 < M < 27

   1. Initial program 75.4%

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

   3. Taylor expanded in l around inf 36.8%

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

      1. mul-1-neg 36.8%

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

   5. Simplified 36.8%

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

   6. Taylor expanded in K around 0 42.7%

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

      1. exp-neg 42.7%

         \[\leadsto \cos \left(-M\right) \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

      2. associate-*r/ 42.7%

         \[\leadsto \color{blue}{\frac{\cos \left(-M\right) \cdot 1}{e^{\ell}}} \]

      3. *-rgt-identity 42.7%

         \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]

      4. cos-neg 42.7%

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

   8. Simplified 42.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -26 \lor \neg \left(M \leq 27\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -680:\\ \;\;\;\;e^{\ell} \cdot \cos \left(\left(m + n\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -680.0)
   (* (exp l) (cos (- (* (+ m n) (* 0.5 K)) M)))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -680.0) {
		tmp = exp(l) * cos((((m + n) * (0.5 * K)) - M));
	} else {
		tmp = cos(M) / exp(l);
	}
	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 (l <= (-680.0d0)) then
        tmp = exp(l) * cos((((m + n) * (0.5d0 * k)) - m_1))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -680.0) {
		tmp = Math.exp(l) * Math.cos((((m + n) * (0.5 * K)) - M));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -680.0:
		tmp = math.exp(l) * math.cos((((m + n) * (0.5 * K)) - M))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -680.0)
		tmp = Float64(exp(l) * cos(Float64(Float64(Float64(m + n) * Float64(0.5 * K)) - M)));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -680.0)
		tmp = exp(l) * cos((((m + n) * (0.5 * K)) - M));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -680.0], N[(N[Exp[l], $MachinePrecision] * N[Cos[N[(N[(N[(m + n), $MachinePrecision] * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -680

   1. Initial program 85.7%

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

   3. Taylor expanded in l around inf 26.7%

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

      1. mul-1-neg 26.7%

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

   5. Simplified 26.7%

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

      1. pow1 26.7%

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

      2. expm1-log1p-u 19.2%

         \[\leadsto {\left(\cos \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(K \cdot \left(m + n\right)\right)\right)}}{2} - M\right) \cdot e^{-\ell}\right)}^{1} \]

      3. expm1-log1p-u 26.7%

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

      4. associate-/l* 26.7%

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

      5. fma-neg 26.7%

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

      6. div-inv 26.7%

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

      7. metadata-eval 26.7%

         \[\leadsto {\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot \color{blue}{0.5}, -M\right)\right) \cdot e^{-\ell}\right)}^{1} \]

      8. add-sqr-sqrt 26.7%

         \[\leadsto {\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right) \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)}^{1} \]

      9. sqrt-unprod 26.7%

         \[\leadsto {\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right) \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)}^{1} \]

      10. sqr-neg 26.7%

          \[\leadsto {\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right) \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)}^{1} \]

      11. sqrt-unprod 0.0%

          \[\leadsto {\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right) \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{1} \]

      12. add-sqr-sqrt 60.4%

          \[\leadsto {\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right) \cdot e^{\color{blue}{\ell}}\right)}^{1} \]

   7. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{{\left(\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right) \cdot e^{\ell}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 60.4%

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

      2. *-commutative 60.4%

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

      3. fma-neg 60.4%

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

      4. associate-*r* 60.4%

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

      5. *-commutative 60.4%

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

      6. associate-*r* 60.4%

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

      7. fma-neg 60.4%

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

      8. +-commutative 60.4%

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

      9. remove-double-neg 60.4%

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

      10. sub-neg 60.4%

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

      11. neg-mul-1 60.4%

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

      12. fma-neg 60.4%

          \[\leadsto e^{\ell} \cdot \cos \color{blue}{\left(\left(0.5 \cdot K\right) \cdot \left(n - -1 \cdot m\right) - M\right)} \]

      13. *-commutative 60.4%

          \[\leadsto e^{\ell} \cdot \cos \left(\color{blue}{\left(K \cdot 0.5\right)} \cdot \left(n - -1 \cdot m\right) - M\right) \]

      14. sub-neg 60.4%

          \[\leadsto e^{\ell} \cdot \cos \left(\left(K \cdot 0.5\right) \cdot \color{blue}{\left(n + \left(--1 \cdot m\right)\right)} - M\right) \]

      15. neg-mul-1 60.4%

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

      16. remove-double-neg 60.4%

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

      17. +-commutative 60.4%

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

   9. Simplified 60.4%

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

   if -680 < l

   1. Initial program 75.1%

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

   3. Taylor expanded in l around inf 32.1%

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

      1. mul-1-neg 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in K around 0 39.9%

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

      1. exp-neg 39.9%

         \[\leadsto \cos \left(-M\right) \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

      2. associate-*r/ 39.9%

         \[\leadsto \color{blue}{\frac{\cos \left(-M\right) \cdot 1}{e^{\ell}}} \]

      3. *-rgt-identity 39.9%

         \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]

      4. cos-neg 39.9%

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

   8. Simplified 39.9%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -680:\\ \;\;\;\;e^{\ell} \cdot \cos \left(\left(m + n\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\cos M}{e^{\ell}} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (/ (cos M) (exp l)))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / 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 = cos(m_1) / exp(l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) / Math.exp(l);
}

Python

def code(K, m, n, M, l):
	return math.cos(M) / math.exp(l)

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) / exp(l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) / exp(l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in l around inf 30.6%

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

   1. mul-1-neg 30.6%

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

5. Simplified 30.6%

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

6. Taylor expanded in K around 0 35.9%

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

   1. exp-neg 35.9%

      \[\leadsto \cos \left(-M\right) \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

   2. associate-*r/ 35.9%

      \[\leadsto \color{blue}{\frac{\cos \left(-M\right) \cdot 1}{e^{\ell}}} \]

   3. *-rgt-identity 35.9%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]

   4. cos-neg 35.9%

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

8. Simplified 35.9%

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

9. Final simplification 35.9%

   \[\leadsto \frac{\cos M}{e^{\ell}} \]
10. Add Preprocessing

Alternative 9: 7.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (cos M))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M);
}

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(m_1)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M);
}

Python

def code(K, m, n, M, l):
	return math.cos(M)

Julia

function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in l around inf 30.6%

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

   1. mul-1-neg 30.6%

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

5. Simplified 30.6%

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

6. Taylor expanded in l around 0 5.9%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right)} \]
7. Step-by-step derivation

   1. associate-*r* 5.9%

      \[\leadsto \cos \left(\color{blue}{\left(0.5 \cdot K\right) \cdot \left(m + n\right)} - M\right) \]

   2. fma-neg 5.9%

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

   3. +-commutative 5.9%

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

   4. remove-double-neg 5.9%

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

   5. sub-neg 5.9%

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

   6. neg-mul-1 5.9%

      \[\leadsto \cos \left(\mathsf{fma}\left(0.5 \cdot K, n - \color{blue}{-1 \cdot m}, -M\right)\right) \]

   7. fma-neg 5.9%

      \[\leadsto \cos \color{blue}{\left(\left(0.5 \cdot K\right) \cdot \left(n - -1 \cdot m\right) - M\right)} \]

   8. *-commutative 5.9%

      \[\leadsto \cos \left(\color{blue}{\left(K \cdot 0.5\right)} \cdot \left(n - -1 \cdot m\right) - M\right) \]

   9. sub-neg 5.9%

      \[\leadsto \cos \left(\left(K \cdot 0.5\right) \cdot \color{blue}{\left(n + \left(--1 \cdot m\right)\right)} - M\right) \]

   10. neg-mul-1 5.9%

       \[\leadsto \cos \left(\left(K \cdot 0.5\right) \cdot \left(n + \left(-\color{blue}{\left(-m\right)}\right)\right) - M\right) \]

   11. remove-double-neg 5.9%

       \[\leadsto \cos \left(\left(K \cdot 0.5\right) \cdot \left(n + \color{blue}{m}\right) - M\right) \]

   12. +-commutative 5.9%

       \[\leadsto \cos \left(\left(K \cdot 0.5\right) \cdot \color{blue}{\left(m + n\right)} - M\right) \]

8. Simplified 5.9%

   \[\leadsto \color{blue}{\cos \left(\left(K \cdot 0.5\right) \cdot \left(m + n\right) - M\right)} \]

9. Taylor expanded in K around 0 6.4%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \]
10. Step-by-step derivation

    1. cos-neg 6.4%

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

11. Simplified 6.4%

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

12. Final simplification 6.4%

    \[\leadsto \cos M \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(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)))))))