Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.9%

Time: 14.6s

Alternatives: 9

Speedup: 3.7×

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: 75.9% 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

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- m n)) 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((m - n)) - 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((m - n)) - 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((m - n)) - 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((m - n)) - 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(m - n)) - 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((m - n)) - 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[(m - n), $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 71.5%

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

3. Taylor expanded in K around 0 94.4%

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

   1. cos-neg 94.4%

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

5. Simplified 94.4%

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

6. Final simplification 94.4%

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

Alternative 2: 95.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -3.2 \cdot 10^{+17} \lor \neg \left(M \leq 3 \cdot 10^{+16}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -3.2e+17) (not (<= M 3e+16)))
   (* (cos M) (exp (- (pow M 2.0))))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (* (+ m n) (+ m n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -3.2e+17) || !(M <= 3e+16)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * ((m + n) * (m + n))))));
	}
	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 <= (-3.2d+17)) .or. (.not. (m_1 <= 3d+16))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * ((m + n) * (m + n))))))
    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 <= -3.2e+17) || !(M <= 3e+16)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * ((m + n) * (m + n))))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -3.2e+17) or not (M <= 3e+16):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * ((m + n) * (m + n))))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -3.2e+17) || !(M <= 3e+16))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -3.2e+17) || ~((M <= 3e+16)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * ((m + n) * (m + n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -3.2e+17], N[Not[LessEqual[M, 3e+16]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.2e17 or 3e16 < M

   1. Initial program 77.5%

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

   3. Taylor expanded in K around 0 99.1%

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

      1. cos-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in M around inf 97.3%

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

      1. mul-1-neg 97.3%

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

   8. Simplified 97.3%

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

   if -3.2e17 < M < 3e16

   1. Initial program 66.9%

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

   3. Taylor expanded in K around 0 90.7%

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

      1. cos-neg 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in M around 0 90.7%

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

      1. unpow2 90.7%

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

      2. +-commutative 90.7%

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

      3. +-commutative 90.7%

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

   8. Applied egg-rr 90.7%

      \[\leadsto e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \color{blue}{\left(\left(n + m\right) \cdot \left(n + m\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.2 \cdot 10^{+17} \lor \neg \left(M \leq 3 \cdot 10^{+16}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((m - n)) - (l + (0.25 * ((m + n) * (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 = exp((abs((m - n)) - (l + (0.25d0 * ((m + n) * (m + n))))))
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 * ((m + n) * (m + n))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in K around 0 94.4%

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

   1. cos-neg 94.4%

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

5. Simplified 94.4%

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

6. Taylor expanded in M around 0 82.9%

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

   1. unpow2 82.9%

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

   2. +-commutative 82.9%

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

   3. +-commutative 82.9%

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

8. Applied egg-rr 82.9%

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

9. Final simplification 82.9%

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

Alternative 4: 87.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in K around 0 94.4%

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

   1. cos-neg 94.4%

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

5. Simplified 94.4%

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

6. Taylor expanded in M around 0 82.9%

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

   1. unpow2 82.9%

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

   2. +-commutative 82.9%

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

   3. +-commutative 82.9%

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

8. Applied egg-rr 82.9%

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

9. Taylor expanded in m around 0 82.7%

   \[\leadsto e^{\left|\color{blue}{-1 \cdot n}\right| - \left(\ell + 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)\right)} \]
10. Step-by-step derivation

    1. neg-mul-1 82.7%

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

11. Simplified 82.7%

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

12. Final simplification 82.7%

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

Alternative 5: 87.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in K around 0 94.4%

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

   1. cos-neg 94.4%

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

5. Simplified 94.4%

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

6. Taylor expanded in M around 0 82.9%

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

   1. unpow2 82.9%

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

   2. +-commutative 82.9%

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

   3. +-commutative 82.9%

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

8. Applied egg-rr 82.9%

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

9. Taylor expanded in m around inf 82.7%

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

10. Final simplification 82.7%

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

Alternative 6: 87.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in K around 0 94.4%

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

   1. cos-neg 94.4%

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

5. Simplified 94.4%

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

6. Taylor expanded in M around 0 82.9%

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

   1. unpow2 82.9%

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

   2. +-commutative 82.9%

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

   3. +-commutative 82.9%

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

8. Applied egg-rr 82.9%

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

9. Taylor expanded in m around -inf 82.9%

   \[\leadsto e^{\color{blue}{\left|-\left(n + -1 \cdot m\right)\right|} - \left(\ell + 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)\right)} \]
10. Step-by-step derivation

    1. fabs-neg 82.9%

       \[\leadsto e^{\color{blue}{\left|n + -1 \cdot m\right|} - \left(\ell + 0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)\right)} \]

    2. mul-1-neg 82.9%

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

    3. sub-neg 82.9%

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

    4. fabs-sub 82.9%

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

    5. rem-square-sqrt 46.0%

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

    6. fabs-sqr 46.0%

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

    7. rem-square-sqrt 82.5%

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

11. Simplified 82.5%

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

12. Final simplification 82.5%

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

Alternative 7: 35.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in l around inf 28.8%

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

   1. mul-1-neg 28.8%

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

5. Simplified 28.8%

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

6. Taylor expanded in n around inf 27.1%

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

   1. associate-/l* 25.4%

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

8. Simplified 25.4%

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

9. Taylor expanded in K around inf 25.6%

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

    1. associate-*r* 25.9%

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

    2. *-commutative 25.9%

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

11. Simplified 25.9%

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

12. Taylor expanded in K around 0 35.4%

    \[\leadsto \color{blue}{e^{-\ell}} \]
13. Add Preprocessing

Alternative 8: 7.0% 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 71.5%

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

3. Taylor expanded in l around inf 28.8%

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

   1. mul-1-neg 28.8%

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

5. Simplified 28.8%

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

6. Taylor expanded in l around 0 6.7%

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

7. Taylor expanded in K around 0 7.3%

   \[\leadsto \cos \color{blue}{\left(-1 \cdot M\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 7.3%

      \[\leadsto \cos \color{blue}{\left(-M\right)} \]

9. Simplified 7.3%

   \[\leadsto \cos \color{blue}{\left(-M\right)} \]

10. Taylor expanded in M around -inf 7.3%

    \[\leadsto \color{blue}{\cos \left(-1 \cdot M\right)} \]
11. Step-by-step derivation

    1. neg-mul-1 7.3%

       \[\leadsto \cos \color{blue}{\left(-M\right)} \]

    2. cos-neg 7.3%

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

12. Simplified 7.3%

    \[\leadsto \color{blue}{\cos M} \]
13. Add Preprocessing

Alternative 9: 7.0% accurate, 425.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

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 = 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in l around inf 28.8%

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

   1. mul-1-neg 28.8%

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

5. Simplified 28.8%

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

6. Taylor expanded in l around 0 6.7%

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

7. Taylor expanded in K around 0 7.3%

   \[\leadsto \cos \color{blue}{\left(-1 \cdot M\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 7.3%

      \[\leadsto \cos \color{blue}{\left(-M\right)} \]

9. Simplified 7.3%

   \[\leadsto \cos \color{blue}{\left(-M\right)} \]

10. Taylor expanded in M around 0 7.3%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))))))