Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.7%

Time: 30.5s

Alternatives: 6

Speedup: 2.0×

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: 76.4% 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: 96.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* (+ m n) 0.5) M)))
   (* (cos M) (exp (- (fabs (- m n)) (+ (* 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((m - n)) - ((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((m - n)) - ((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((m - n)) - ((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((m - n)) - ((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(m - n)) - 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((m - n)) - ((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[(m - n), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$0 * t$95$0), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.2%

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

3. Simplified 77.6%

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

4. Taylor expanded in K around 0 97.1%

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

   1. cos-neg 97.1%

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

6. Simplified 97.1%

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

   1. unpow2 97.1%

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

   2. div-inv 97.1%

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

   3. metadata-eval 97.1%

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

   4. div-inv 97.1%

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

   5. metadata-eval 97.1%

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

8. Applied egg-rr 97.1%

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

9. Final simplification 97.1%

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

Alternative 2: 96.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((m - n)) - l) - pow((((m + n) * 0.5) - 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 = exp(((abs((m - n)) - l) - ((((m + n) * 0.5d0) - m_1) ** 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) - Math.pow((((m + n) * 0.5) - M), 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Simplified 77.6%

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

4. Taylor expanded in M around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. *-commutative 76.9%

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

   3. associate-*r* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in K around 0 96.8%

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

   1. associate--r+ 96.8%

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

   2. *-commutative 96.8%

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

   3. sub-neg 96.8%

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

   4. mul-1-neg 96.8%

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

   5. +-commutative 96.8%

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

   6. *-commutative 96.8%

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

9. Simplified 96.8%

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

10. Final simplification 96.8%

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

Alternative 3: 86.5% 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 77.2%

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

3. Simplified 77.6%

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

4. Taylor expanded in K around 0 86.0%

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

   1. cos-neg 86.0%

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

   2. associate-*r* 86.0%

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

   3. *-commutative 86.0%

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

   4. sin-neg 86.0%

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

6. Simplified 86.0%

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

7. Taylor expanded in M around 0 88.8%

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

   1. +-commutative 88.8%

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

   2. +-commutative 88.8%

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

9. Simplified 88.8%

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

    1. unpow2 88.8%

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

    2. +-commutative 88.8%

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

    3. +-commutative 88.8%

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

11. Applied egg-rr 88.8%

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

12. Final simplification 88.8%

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

Alternative 4: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot 0.5 - M\\ t_1 := n \cdot 0.5 - M\\ t_2 := e^{\left(n + t_1 \cdot \left(m + t_1\right)\right) - \left(m + \ell\right)}\\ \mathbf{if}\;n \leq -1.08 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 4:\\ \;\;\;\;e^{\left(n + 0.25 \cdot {n}^{2}\right) - \left(m + \ell\right)}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+140}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n + t_0 \cdot \left(n + t_0\right)\right) - \left(m + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* m 0.5) M))
        (t_1 (- (* n 0.5) M))
        (t_2 (exp (- (+ n (* t_1 (+ m t_1))) (+ m l)))))
   (if (<= n -1.08e-184)
     t_2
     (if (<= n 4.0)
       (exp (- (+ n (* 0.25 (pow n 2.0))) (+ m l)))
       (if (<= n 2e+140) t_2 (exp (- (+ n (* t_0 (+ n t_0))) (+ m l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double t_1 = (n * 0.5) - M;
	double t_2 = exp(((n + (t_1 * (m + t_1))) - (m + l)));
	double tmp;
	if (n <= -1.08e-184) {
		tmp = t_2;
	} else if (n <= 4.0) {
		tmp = exp(((n + (0.25 * pow(n, 2.0))) - (m + l)));
	} else if (n <= 2e+140) {
		tmp = t_2;
	} else {
		tmp = exp(((n + (t_0 * (n + t_0))) - (m + 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (m * 0.5d0) - m_1
    t_1 = (n * 0.5d0) - m_1
    t_2 = exp(((n + (t_1 * (m + t_1))) - (m + l)))
    if (n <= (-1.08d-184)) then
        tmp = t_2
    else if (n <= 4.0d0) then
        tmp = exp(((n + (0.25d0 * (n ** 2.0d0))) - (m + l)))
    else if (n <= 2d+140) then
        tmp = t_2
    else
        tmp = exp(((n + (t_0 * (n + t_0))) - (m + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double t_1 = (n * 0.5) - M;
	double t_2 = Math.exp(((n + (t_1 * (m + t_1))) - (m + l)));
	double tmp;
	if (n <= -1.08e-184) {
		tmp = t_2;
	} else if (n <= 4.0) {
		tmp = Math.exp(((n + (0.25 * Math.pow(n, 2.0))) - (m + l)));
	} else if (n <= 2e+140) {
		tmp = t_2;
	} else {
		tmp = Math.exp(((n + (t_0 * (n + t_0))) - (m + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m * 0.5) - M
	t_1 = (n * 0.5) - M
	t_2 = math.exp(((n + (t_1 * (m + t_1))) - (m + l)))
	tmp = 0
	if n <= -1.08e-184:
		tmp = t_2
	elif n <= 4.0:
		tmp = math.exp(((n + (0.25 * math.pow(n, 2.0))) - (m + l)))
	elif n <= 2e+140:
		tmp = t_2
	else:
		tmp = math.exp(((n + (t_0 * (n + t_0))) - (m + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m * 0.5) - M)
	t_1 = Float64(Float64(n * 0.5) - M)
	t_2 = exp(Float64(Float64(n + Float64(t_1 * Float64(m + t_1))) - Float64(m + l)))
	tmp = 0.0
	if (n <= -1.08e-184)
		tmp = t_2;
	elseif (n <= 4.0)
		tmp = exp(Float64(Float64(n + Float64(0.25 * (n ^ 2.0))) - Float64(m + l)));
	elseif (n <= 2e+140)
		tmp = t_2;
	else
		tmp = exp(Float64(Float64(n + Float64(t_0 * Float64(n + t_0))) - Float64(m + l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m * 0.5) - M;
	t_1 = (n * 0.5) - M;
	t_2 = exp(((n + (t_1 * (m + t_1))) - (m + l)));
	tmp = 0.0;
	if (n <= -1.08e-184)
		tmp = t_2;
	elseif (n <= 4.0)
		tmp = exp(((n + (0.25 * (n ^ 2.0))) - (m + l)));
	elseif (n <= 2e+140)
		tmp = t_2;
	else
		tmp = exp(((n + (t_0 * (n + t_0))) - (m + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(n + N[(t$95$1 * N[(m + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.08e-184], t$95$2, If[LessEqual[n, 4.0], N[Exp[N[(N[(n + N[(0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2e+140], t$95$2, N[Exp[N[(N[(n + N[(t$95$0 * N[(n + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.07999999999999995e-184 or 4 < n < 2.00000000000000012e140

   1. Initial program 77.8%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in M around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*r* 77.0%

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

   6. Simplified 77.0%

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

      1. log1p-expm1-u 77.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}\right)\right)} \]

      2. associate-+l- 77.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}}\right)\right) \]

   8. Applied egg-rr 9.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\ell - \left(n - m\right)\right)}\right)\right)} \]

   9. Taylor expanded in K around 0 9.8%

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

   10. Taylor expanded in m around 0 24.2%

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

       1. +-commutative 24.2%

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

       2. unpow2 24.2%

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

       3. distribute-rgt-out 27.4%

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

   12. Simplified 27.4%

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

   if -1.07999999999999995e-184 < n < 4

   1. Initial program 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 81.3%

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

      1. *-commutative 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*r* 81.3%

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

   6. Simplified 81.3%

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

      1. log1p-expm1-u 81.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}\right)\right)} \]

      2. associate-+l- 81.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}}\right)\right) \]

   8. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\ell - \left(n - m\right)\right)}\right)\right)} \]

   9. Taylor expanded in K around 0 24.6%

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

   10. Taylor expanded in n around inf 52.6%

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

   if 2.00000000000000012e140 < n

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in M around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. *-commutative 69.4%

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

      3. associate-*r* 69.4%

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

   6. Simplified 69.4%

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

      1. log1p-expm1-u 69.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}\right)\right)} \]

      2. associate-+l- 69.4%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}}\right)\right) \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\ell - \left(n - m\right)\right)}\right)\right)} \]

   9. Taylor expanded in K around 0 1.6%

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

   10. Taylor expanded in n around 0 17.2%

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

       1. +-commutative 17.2%

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

       2. unpow2 17.2%

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

       3. distribute-rgt-out 33.7%

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

   12. Simplified 33.7%

       \[\leadsto e^{\left(n + \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell + m\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.08 \cdot 10^{-184}:\\ \;\;\;\;e^{\left(n + \left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right)\right) - \left(m + \ell\right)}\\ \mathbf{elif}\;n \leq 4:\\ \;\;\;\;e^{\left(n + 0.25 \cdot {n}^{2}\right) - \left(m + \ell\right)}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+140}:\\ \;\;\;\;e^{\left(n + \left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right)\right) - \left(m + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n + \left(m \cdot 0.5 - M\right) \cdot \left(n + \left(m \cdot 0.5 - M\right)\right)\right) - \left(m + \ell\right)}\\ \end{array} \]

Alternative 5: 32.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot 0.5 - M\\ t_1 := n \cdot 0.5 - M\\ \mathbf{if}\;n \leq 10^{+134}:\\ \;\;\;\;e^{\left(n + t_1 \cdot \left(m + t_1\right)\right) - \left(m + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n + t_0 \cdot \left(n + t_0\right)\right) - \left(m + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* m 0.5) M)) (t_1 (- (* n 0.5) M)))
   (if (<= n 1e+134)
     (exp (- (+ n (* t_1 (+ m t_1))) (+ m l)))
     (exp (- (+ n (* t_0 (+ n t_0))) (+ m l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double t_1 = (n * 0.5) - M;
	double tmp;
	if (n <= 1e+134) {
		tmp = exp(((n + (t_1 * (m + t_1))) - (m + l)));
	} else {
		tmp = exp(((n + (t_0 * (n + t_0))) - (m + 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * 0.5d0) - m_1
    t_1 = (n * 0.5d0) - m_1
    if (n <= 1d+134) then
        tmp = exp(((n + (t_1 * (m + t_1))) - (m + l)))
    else
        tmp = exp(((n + (t_0 * (n + t_0))) - (m + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double t_1 = (n * 0.5) - M;
	double tmp;
	if (n <= 1e+134) {
		tmp = Math.exp(((n + (t_1 * (m + t_1))) - (m + l)));
	} else {
		tmp = Math.exp(((n + (t_0 * (n + t_0))) - (m + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m * 0.5) - M
	t_1 = (n * 0.5) - M
	tmp = 0
	if n <= 1e+134:
		tmp = math.exp(((n + (t_1 * (m + t_1))) - (m + l)))
	else:
		tmp = math.exp(((n + (t_0 * (n + t_0))) - (m + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m * 0.5) - M)
	t_1 = Float64(Float64(n * 0.5) - M)
	tmp = 0.0
	if (n <= 1e+134)
		tmp = exp(Float64(Float64(n + Float64(t_1 * Float64(m + t_1))) - Float64(m + l)));
	else
		tmp = exp(Float64(Float64(n + Float64(t_0 * Float64(n + t_0))) - Float64(m + l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m * 0.5) - M;
	t_1 = (n * 0.5) - M;
	tmp = 0.0;
	if (n <= 1e+134)
		tmp = exp(((n + (t_1 * (m + t_1))) - (m + l)));
	else
		tmp = exp(((n + (t_0 * (n + t_0))) - (m + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[n, 1e+134], N[Exp[N[(N[(n + N[(t$95$1 * N[(m + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n + N[(t$95$0 * N[(n + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if n < 9.99999999999999921e133

   1. Initial program 79.1%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in M around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*r* 78.6%

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

   6. Simplified 78.6%

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

      1. log1p-expm1-u 78.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}\right)\right)} \]

      2. associate-+l- 78.6%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}}\right)\right) \]

   8. Applied egg-rr 15.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\ell - \left(n - m\right)\right)}\right)\right)} \]

   9. Taylor expanded in K around 0 15.4%

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

   10. Taylor expanded in m around 0 30.5%

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

       1. +-commutative 30.5%

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

       2. unpow2 30.5%

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

       3. distribute-rgt-out 32.5%

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

   12. Simplified 32.5%

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

   if 9.99999999999999921e133 < n

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in M around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. *-commutative 69.4%

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

      3. associate-*r* 69.4%

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

   6. Simplified 69.4%

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

      1. log1p-expm1-u 69.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}\right)\right)} \]

      2. associate-+l- 69.4%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}}\right)\right) \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\ell - \left(n - m\right)\right)}\right)\right)} \]

   9. Taylor expanded in K around 0 1.6%

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

   10. Taylor expanded in n around 0 17.2%

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

       1. +-commutative 17.2%

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

       2. unpow2 17.2%

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

       3. distribute-rgt-out 33.7%

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

   12. Simplified 33.7%

       \[\leadsto e^{\left(n + \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell + m\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{+134}:\\ \;\;\;\;e^{\left(n + \left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right)\right) - \left(m + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n + \left(m \cdot 0.5 - M\right) \cdot \left(n + \left(m \cdot 0.5 - M\right)\right)\right) - \left(m + \ell\right)}\\ \end{array} \]

Alternative 6: 29.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Simplified 77.6%

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

4. Taylor expanded in M around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. *-commutative 76.9%

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

   3. associate-*r* 76.9%

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

6. Simplified 76.9%

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

   1. log1p-expm1-u 76.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}\right)\right)} \]

   2. associate-+l- 76.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{\color{blue}{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|n - m\right|\right)}}\right)\right) \]

8. Applied egg-rr 12.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right)\right) \cdot e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} - \left(\ell - \left(n - m\right)\right)}\right)\right)} \]

9. Taylor expanded in K around 0 12.8%

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

10. Taylor expanded in n around 0 24.5%

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

    1. +-commutative 24.5%

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

    2. unpow2 24.5%

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

    3. distribute-rgt-out 29.7%

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

12. Simplified 29.7%

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

13. Final simplification 29.7%

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

Reproduce

herbie shell --seed 2023336 
(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)))))))