Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.7% → 96.8%

Time: 18.8s

Alternatives: 9

Speedup: 3.8×

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: 76.7% 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: 96.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (/ (cos 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(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(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(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(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(M) / exp(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(M) / exp((((((m + n) / 2.0) - M) ^ 2.0) + (l - abs((m - n)))));
end

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

6. Simplified 98.0%

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

7. Final simplification 98.0%

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

Alternative 2: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -4 \cdot 10^{+24} \lor \neg \left(M \leq 7.8 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right) - \left|m - n\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -4e+24) (not (<= M 7.8e+95)))
   (/ (cos M) (exp (* M M)))
   (/ 1.0 (exp (- (+ l (* 0.25 (* (+ m n) (+ m n)))) (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -4e+24) || !(M <= 7.8e+95)) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = 1.0 / exp(((l + (0.25 * ((m + n) * (m + n)))) - fabs((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 <= (-4d+24)) .or. (.not. (m_1 <= 7.8d+95))) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = 1.0d0 / exp(((l + (0.25d0 * ((m + n) * (m + n)))) - abs((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 <= -4e+24) || !(M <= 7.8e+95)) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = 1.0 / Math.exp(((l + (0.25 * ((m + n) * (m + n)))) - Math.abs((m - n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -4e+24) or not (M <= 7.8e+95):
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = 1.0 / math.exp(((l + (0.25 * ((m + n) * (m + n)))) - math.fabs((m - n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -4e+24) || !(M <= 7.8e+95))
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(1.0 / exp(Float64(Float64(l + Float64(0.25 * Float64(Float64(m + n) * Float64(m + n)))) - abs(Float64(m - n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -4e+24) || ~((M <= 7.8e+95)))
		tmp = cos(M) / exp((M * M));
	else
		tmp = 1.0 / exp(((l + (0.25 * ((m + n) * (m + n)))) - abs((m - n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -4e+24], N[Not[LessEqual[M, 7.8e+95]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(N[(l + N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.9999999999999999e24 or 7.7999999999999994e95 < M

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 99.0%

      \[\leadsto \frac{\cos M}{e^{\color{blue}{{M}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 99.0%

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

   9. Simplified 99.0%

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

   if -3.9999999999999999e24 < M < 7.7999999999999994e95

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in M around 0 94.9%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]
   8. Step-by-step derivation

      1. unpow2 94.9%

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

   9. Applied egg-rr 94.9%

      \[\leadsto \frac{1}{e^{\left(0.25 \cdot \color{blue}{\left(\left(n + m\right) \cdot \left(n + m\right)\right)} + \ell\right) - \left|n - m\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4 \cdot 10^{+24} \lor \neg \left(M \leq 7.8 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\left(\ell + 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right) - \left|m - n\right|}}\\ \end{array} \]

Alternative 3: 73.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2400:\\ \;\;\;\;e^{\left(\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2400.0)
   (exp (- (- (fabs (- m n)) (* 0.25 (* m m))) l))
   (/ 1.0 (exp (* 0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2400.0) {
		tmp = exp(((fabs((m - n)) - (0.25 * (m * m))) - l));
	} else {
		tmp = 1.0 / exp((0.25 * (n * 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 (n <= 2400.0d0) then
        tmp = exp(((abs((m - n)) - (0.25d0 * (m * m))) - l))
    else
        tmp = 1.0d0 / exp((0.25d0 * (n * 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 (n <= 2400.0) {
		tmp = Math.exp(((Math.abs((m - n)) - (0.25 * (m * m))) - l));
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2400.0:
		tmp = math.exp(((math.fabs((m - n)) - (0.25 * (m * m))) - l))
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2400.0)
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - Float64(0.25 * Float64(m * m))) - l));
	else
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2400.0)
		tmp = exp(((abs((m - n)) - (0.25 * (m * m))) - l));
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2400.0], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2400

   1. Initial program 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in M around 0 88.6%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]

   8. Taylor expanded in n around 0 70.3%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(\ell + 0.25 \cdot {m}^{2}\right) - \left|n - m\right|}}} \]
   9. Step-by-step derivation

      1. rec-exp 70.3%

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

      2. associate--l+ 70.3%

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

      3. *-commutative 70.3%

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

      4. unpow2 70.3%

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

   10. Simplified 70.3%

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

   if 2400 < n

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]

   8. Taylor expanded in n around inf 100.0%

      \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot {n}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{{n}^{2} \cdot 0.25}}} \]

      2. unpow2 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{\left(n \cdot n\right) \cdot 0.25}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2400:\\ \;\;\;\;e^{\left(\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 4: 77.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1550:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell + \left(0.25 \cdot \left(n \cdot n\right) - n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1550.0)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (/ 1.0 (exp (+ l (- (* 0.25 (* n n)) n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1550.0) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else {
		tmp = 1.0 / exp((l + ((0.25 * (n * n)) - 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 <= (-1550.0d0)) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else
        tmp = 1.0d0 / exp((l + ((0.25d0 * (n * n)) - 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 <= -1550.0) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else {
		tmp = 1.0 / Math.exp((l + ((0.25 * (n * n)) - n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1550.0:
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	else:
		tmp = 1.0 / math.exp((l + ((0.25 * (n * n)) - n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1550.0)
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	else
		tmp = Float64(1.0 / exp(Float64(l + Float64(Float64(0.25 * Float64(n * n)) - n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1550.0)
		tmp = 1.0 / exp((0.25 * (m * m)));
	else
		tmp = 1.0 / exp((l + ((0.25 * (n * n)) - n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1550.0], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(l + N[(N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1550

   1. Initial program 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in M around 0 98.8%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]

   8. Taylor expanded in m around inf 96.3%

      \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot {m}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto \frac{1}{e^{\color{blue}{{m}^{2} \cdot 0.25}}} \]

      2. unpow2 96.3%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25}} \]

   10. Simplified 96.3%

       \[\leadsto \frac{1}{e^{\color{blue}{\left(m \cdot m\right) \cdot 0.25}}} \]

   if -1550 < m

   1. Initial program 78.1%

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

   3. Simplified 78.1%

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

      1. add-cbrt-cube 78.1%

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

      2. pow3 78.1%

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

   5. Applied egg-rr 77.6%

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

   6. Taylor expanded in m around 0 77.2%

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

   7. Taylor expanded in M around 0 62.8%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot \left(n \cdot K\right)\right)}{e^{\left(\ell + 0.25 \cdot {n}^{2}\right) - n}}} \]
   8. Step-by-step derivation

      1. associate--l+ 62.8%

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

      2. *-commutative 62.8%

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

      3. unpow2 62.8%

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

   9. Simplified 62.8%

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

   10. Taylor expanded in n around 0 77.8%

       \[\leadsto \frac{\color{blue}{1}}{e^{\ell + \left(\left(n \cdot n\right) \cdot 0.25 - n\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1550:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell + \left(0.25 \cdot \left(n \cdot n\right) - n\right)}}\\ \end{array} \]

Alternative 5: 69.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00015 \lor \neg \left(m \leq 52\right):\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -0.00015) (not (<= m 52.0)))
   (/ 1.0 (exp (* 0.25 (* m m))))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -0.00015) || !(m <= 52.0)) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else {
		tmp = 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 <= (-0.00015d0)) .or. (.not. (m <= 52.0d0))) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else
        tmp = 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 <= -0.00015) || !(m <= 52.0)) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -0.00015) or not (m <= 52.0):
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -0.00015) || !(m <= 52.0))
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -0.00015) || ~((m <= 52.0)))
		tmp = 1.0 / exp((0.25 * (m * m)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -0.00015], N[Not[LessEqual[m, 52.0]], $MachinePrecision]], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.49999999999999987e-4 or 52 < m

   1. Initial program 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in M around 0 97.7%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]

   8. Taylor expanded in m around inf 96.2%

      \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot {m}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 96.2%

         \[\leadsto \frac{1}{e^{\color{blue}{{m}^{2} \cdot 0.25}}} \]

      2. unpow2 96.2%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25}} \]

   10. Simplified 96.2%

       \[\leadsto \frac{1}{e^{\color{blue}{\left(m \cdot m\right) \cdot 0.25}}} \]

   if -1.49999999999999987e-4 < m < 52

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in l around inf 47.2%

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

   8. Taylor expanded in M around 0 46.4%

      \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   9. Step-by-step derivation

      1. rec-exp 46.4%

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

   10. Simplified 46.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00015 \lor \neg \left(m \leq 52\right):\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 6: 62.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2.2e-231)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (if (<= n 54.0) (exp (- l)) (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.2e-231) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = exp(-l);
	} else {
		tmp = 1.0 / exp((0.25 * (n * 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 (n <= 2.2d-231) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else if (n <= 54.0d0) then
        tmp = exp(-l)
    else
        tmp = 1.0d0 / exp((0.25d0 * (n * 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 (n <= 2.2e-231) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = Math.exp(-l);
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.2e-231:
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	elif n <= 54.0:
		tmp = math.exp(-l)
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2.2e-231)
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 54.0)
		tmp = exp(Float64(-l));
	else
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2.2e-231)
		tmp = 1.0 / exp((0.25 * (m * m)));
	elseif (n <= 54.0)
		tmp = exp(-l);
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2.2e-231], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[Exp[(-l)], $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.20000000000000009e-231

   1. Initial program 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in K around 0 97.8%

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

      1. cos-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in M around 0 90.9%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]

   8. Taylor expanded in m around inf 56.3%

      \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot {m}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 56.3%

         \[\leadsto \frac{1}{e^{\color{blue}{{m}^{2} \cdot 0.25}}} \]

      2. unpow2 56.3%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25}} \]

   10. Simplified 56.3%

       \[\leadsto \frac{1}{e^{\color{blue}{\left(m \cdot m\right) \cdot 0.25}}} \]

   if 2.20000000000000009e-231 < n < 54

   1. Initial program 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in l around inf 56.5%

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

   8. Taylor expanded in M around 0 56.5%

      \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   9. Step-by-step derivation

      1. rec-exp 56.5%

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

   10. Simplified 56.5%

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

   if 54 < n

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{\left(0.25 \cdot {\left(n + m\right)}^{2} + \ell\right) - \left|n - m\right|}}} \]

   8. Taylor expanded in n around inf 100.0%

      \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot {n}^{2}}}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{{n}^{2} \cdot 0.25}}} \]

      2. unpow2 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(n \cdot n\right)} \cdot 0.25}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{\left(n \cdot n\right) \cdot 0.25}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 7: 35.3% 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 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in l around inf 40.7%

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

8. Taylor expanded in M around 0 40.3%

   \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
9. Step-by-step derivation

   1. rec-exp 40.3%

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

10. Simplified 40.3%

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

11. Final simplification 40.3%

    \[\leadsto e^{-\ell} \]

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 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in l around inf 40.7%

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

8. Taylor expanded in l around 0 7.3%

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

9. Final simplification 7.3%

   \[\leadsto \cos M \]

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 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in K around 0 98.0%

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

   1. cos-neg 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in l around inf 40.7%

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

8. Taylor expanded in l around 0 7.3%

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

9. Taylor expanded in M around 0 7.3%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 7.3%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023195 
(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)))))))