Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.6%

Time: 53.0s

Alternatives: 4

Speedup: 1.4×

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: 75.7% 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: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5 - M\\ \cos M \cdot e^{\left|n - m\right| - \left(t\_0 \cdot t\_0 + \ell\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* (+ m n) 0.5) M)))
   (* (cos M) (exp (- (fabs (- n m)) (+ (* t_0 t_0) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return cos(M) * exp((fabs((n - m)) - ((t_0 * t_0) + 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
    real(8) :: t_0
    t_0 = ((m + n) * 0.5d0) - m_1
    code = cos(m_1) * exp((abs((n - m)) - ((t_0 * t_0) + l)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return Math.cos(M) * Math.exp((Math.abs((n - m)) - ((t_0 * t_0) + l)));
}

Python

def code(K, m, n, M, l):
	t_0 = ((m + n) * 0.5) - M
	return math.cos(M) * math.exp((math.fabs((n - m)) - ((t_0 * t_0) + l)))

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(Float64(m + n) * 0.5) - M)
	return Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64(Float64(t_0 * t_0) + l))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	t_0 = ((m + n) * 0.5) - M;
	tmp = cos(M) * exp((abs((n - m)) - ((t_0 * t_0) + l)));
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision]}, N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$0 * t$95$0), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.4%

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

3. Taylor expanded in K around 0 96.1%

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

   1. cos-neg 96.1%

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

   2. sub-neg 96.1%

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

   3. sub-neg 96.1%

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

   4. +-commutative 96.1%

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

   5. neg-mul-1 96.1%

      \[\leadsto \cos M \cdot e^{\left|m - n\right| + \left(-\left(\ell + {\left(\color{blue}{-1 \cdot M} + 0.5 \cdot \left(m + n\right)\right)}^{2}\right)\right)} \]

   6. sub-neg 96.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(-1 \cdot M + 0.5 \cdot \left(m + n\right)\right)}^{2}\right)}} \]

5. Simplified 96.1%

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

   1. unpow2 96.1%

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

7. Applied egg-rr 96.1%

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

8. Final simplification 96.1%

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

Alternative 2: 33.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 14.5:\\ \;\;\;\;e^{{\left(n \cdot 0.5 - M\right)}^{2} - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{n - 0.25 \cdot {m}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 14.5)
   (exp (- (pow (- (* n 0.5) M) 2.0) (+ n l)))
   (/ (cos M) (exp (- n (* 0.25 (pow m 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 14.5) {
		tmp = exp((pow(((n * 0.5) - M), 2.0) - (n + l)));
	} else {
		tmp = cos(M) / exp((n - (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 (n <= 14.5d0) then
        tmp = exp(((((n * 0.5d0) - m_1) ** 2.0d0) - (n + l)))
    else
        tmp = cos(m_1) / exp((n - (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 (n <= 14.5) {
		tmp = Math.exp((Math.pow(((n * 0.5) - M), 2.0) - (n + l)));
	} else {
		tmp = Math.cos(M) / Math.exp((n - (0.25 * Math.pow(m, 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 14.5:
		tmp = math.exp((math.pow(((n * 0.5) - M), 2.0) - (n + l)))
	else:
		tmp = math.cos(M) / math.exp((n - (0.25 * math.pow(m, 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 14.5)
		tmp = exp(Float64((Float64(Float64(n * 0.5) - M) ^ 2.0) - Float64(n + l)));
	else
		tmp = Float64(cos(M) / exp(Float64(n - Float64(0.25 * (m ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 14.5)
		tmp = exp(((((n * 0.5) - M) ^ 2.0) - (n + l)));
	else
		tmp = cos(M) / exp((n - (0.25 * (m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 14.5], N[Exp[N[(N[Power[N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(n - N[(0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 14.5

   1. Initial program 74.3%

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

      1. add-cube-cbrt 74.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right) \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}} \]

      2. pow3 74.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right)}^{3}} \]

   4. Applied egg-rr 22.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right) \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3}} \]

   5. Taylor expanded in m around inf 21.6%

      \[\leadsto {\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(K \cdot m\right)\right)} \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3} \]
   6. Step-by-step derivation

      1. *-commutative 21.6%

         \[\leadsto {\left(\sqrt[3]{\cos \left(0.5 \cdot \color{blue}{\left(m \cdot K\right)}\right) \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3} \]

   7. Simplified 21.6%

      \[\leadsto {\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(m \cdot K\right)\right)} \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3} \]

   8. Taylor expanded in m around 0 27.9%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot e^{{\left(0.5 \cdot n - M\right)}^{2} - \left(\ell + n\right)}} \]
   9. Step-by-step derivation

      1. pow-base-1 27.9%

         \[\leadsto \color{blue}{1} \cdot e^{{\left(0.5 \cdot n - M\right)}^{2} - \left(\ell + n\right)} \]

      2. *-lft-identity 27.9%

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

      3. *-commutative 27.9%

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

      4. +-commutative 27.9%

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

   10. Simplified 27.9%

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

   if 14.5 < n

   1. Initial program 71.0%

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

      1. exp-diff 8.7%

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

      2. associate-*r/ 8.7%

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

   4. Applied egg-rr 1.7%

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

      1. associate-/l* 1.7%

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

      2. div-exp 6.8%

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

      3. associate--r- 6.8%

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

      4. +-commutative 6.8%

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

   6. Simplified 6.8%

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

   7. Taylor expanded in n around inf 2.5%

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

   8. Taylor expanded in m around inf 48.2%

      \[\leadsto \frac{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}{e^{n - \color{blue}{0.25 \cdot {m}^{2}}}} \]

   9. Taylor expanded in K around 0 68.6%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{n - 0.25 \cdot {m}^{2}}} \]
   10. Step-by-step derivation

       1. cos-neg 68.6%

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

   11. Simplified 68.6%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 14.5:\\ \;\;\;\;e^{{\left(n \cdot 0.5 - M\right)}^{2} - \left(n + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{n - 0.25 \cdot {m}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (- (fabs (- m n)) l) (* 0.25 (pow (+ m n) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((m - n)) - l) - (0.25 * pow((m + n), 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 = exp(((abs((m - n)) - l) - (0.25d0 * ((m + n) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((Math.abs((m - n)) - l) - (0.25 * Math.pow((m + n), 2.0))));
}

Python

def code(K, m, n, M, l):
	return math.exp(((math.fabs((m - n)) - l) - (0.25 * math.pow((m + n), 2.0))))

Julia

function code(K, m, n, M, l)
	return exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(0.25 * (Float64(m + n) ^ 2.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((m - n)) - l) - (0.25 * ((m + n) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.4%

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

3. Taylor expanded in K around 0 96.1%

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

   1. cos-neg 96.1%

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

   2. sub-neg 96.1%

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

   3. sub-neg 96.1%

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

   4. +-commutative 96.1%

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

   5. neg-mul-1 96.1%

      \[\leadsto \cos M \cdot e^{\left|m - n\right| + \left(-\left(\ell + {\left(\color{blue}{-1 \cdot M} + 0.5 \cdot \left(m + n\right)\right)}^{2}\right)\right)} \]

   6. sub-neg 96.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + {\left(-1 \cdot M + 0.5 \cdot \left(m + n\right)\right)}^{2}\right)}} \]

5. Simplified 96.1%

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

6. Taylor expanded in M around 0 86.2%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. associate--r+ 86.2%

      \[\leadsto e^{\color{blue}{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}} \]

   2. fabs-sub 86.2%

      \[\leadsto e^{\left(\color{blue}{\left|n - m\right|} - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}} \]

8. Simplified 86.2%

   \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}}} \]

9. Final simplification 86.2%

   \[\leadsto e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot {\left(m + n\right)}^{2}} \]
10. Add Preprocessing

Alternative 4: 20.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{{\left(n \cdot 0.5 - M\right)}^{2} - \left(n + \ell\right)} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((pow(((n * 0.5) - M), 2.0) - (n + 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 = exp(((((n * 0.5d0) - m_1) ** 2.0d0) - (n + l)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.pow(((n * 0.5) - M), 2.0) - (n + l)));
}

Python

def code(K, m, n, M, l):
	return math.exp((math.pow(((n * 0.5) - M), 2.0) - (n + l)))

Julia

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

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((((n * 0.5) - M) ^ 2.0) - (n + l)));
end

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. add-cube-cbrt 73.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right) \cdot \sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}} \]

   2. pow3 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right)}^{3}} \]

4. Applied egg-rr 18.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right) \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3}} \]

5. Taylor expanded in m around inf 17.3%

   \[\leadsto {\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(K \cdot m\right)\right)} \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3} \]
6. Step-by-step derivation

   1. *-commutative 17.3%

      \[\leadsto {\left(\sqrt[3]{\cos \left(0.5 \cdot \color{blue}{\left(m \cdot K\right)}\right) \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3} \]

7. Simplified 17.3%

   \[\leadsto {\left(\sqrt[3]{\cos \color{blue}{\left(0.5 \cdot \left(m \cdot K\right)\right)} \cdot e^{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \ell\right) + \left(m - n\right)}}\right)}^{3} \]

8. Taylor expanded in m around 0 22.3%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot e^{{\left(0.5 \cdot n - M\right)}^{2} - \left(\ell + n\right)}} \]
9. Step-by-step derivation

   1. pow-base-1 22.3%

      \[\leadsto \color{blue}{1} \cdot e^{{\left(0.5 \cdot n - M\right)}^{2} - \left(\ell + n\right)} \]

   2. *-lft-identity 22.3%

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

   3. *-commutative 22.3%

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

   4. +-commutative 22.3%

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

10. Simplified 22.3%

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

11. Final simplification 22.3%

    \[\leadsto e^{{\left(n \cdot 0.5 - M\right)}^{2} - \left(n + \ell\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))))))