Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.8%

Time: 16.3s

Alternatives: 5

Speedup: 3.5×

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: 76.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0 96.0%

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

5. Simplified 96.0%

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

6. Final simplification 96.0%

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

Alternative 2: 96.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = (0.5d0 * (n + m)) - m_1
    code = exp((abs((n - m)) - ((t_0 * t_0) + l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0 96.0%

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

5. Simplified 96.0%

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

6. Taylor expanded in M around 0 95.3%

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

   1. *-commutative 95.3%

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

   2. metadata-eval 95.3%

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

   3. div-inv 95.3%

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

   4. unpow2 95.3%

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

   5. div-inv 95.3%

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

   6. metadata-eval 95.3%

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

   7. *-commutative 95.3%

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

   8. +-commutative 95.3%

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

   9. div-inv 95.3%

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

   10. metadata-eval 95.3%

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

   11. *-commutative 95.3%

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

   12. +-commutative 95.3%

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

8. Applied egg-rr 95.3%

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

9. Final simplification 95.3%

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

Alternative 3: 96.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in K around 0 96.0%

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

5. Simplified 96.0%

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

6. Taylor expanded in M around 0 95.3%

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

   1. *-commutative 95.3%

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

   2. metadata-eval 95.3%

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

   3. div-inv 95.3%

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

   4. unpow2 95.3%

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

   5. div-inv 95.3%

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

   6. metadata-eval 95.3%

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

   7. *-commutative 95.3%

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

   8. +-commutative 95.3%

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

   9. div-inv 95.3%

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

   10. metadata-eval 95.3%

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

   11. *-commutative 95.3%

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

   12. +-commutative 95.3%

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

8. Applied egg-rr 95.3%

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

9. Taylor expanded in n around -inf 95.3%

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

    1. fabs-neg 95.3%

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

    2. neg-mul-1 95.3%

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

    3. sub-neg 95.3%

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

    4. rem-square-sqrt 48.1%

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

    5. fabs-sqr 48.1%

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

    6. rem-square-sqrt 95.3%

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

11. Simplified 95.3%

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

12. Final simplification 95.3%

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

Alternative 4: 6.9% 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 72.5%

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

3. Taylor expanded in l around inf 30.6%

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

   1. mul-1-neg 30.6%

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

5. Simplified 30.6%

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

6. Taylor expanded in l around 0 7.7%

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

   1. associate-*r* 7.7%

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

8. Simplified 7.7%

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

9. Taylor expanded in K around 0 8.3%

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

    1. cos-neg 8.3%

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

11. Simplified 8.3%

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

Alternative 5: 6.9% 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 72.5%

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

3. Taylor expanded in l around inf 30.6%

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

   1. mul-1-neg 30.6%

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

5. Simplified 30.6%

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

6. Taylor expanded in l around 0 7.7%

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

   1. associate-*r* 7.7%

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

8. Simplified 7.7%

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

9. Taylor expanded in K around 0 7.6%

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

    1. cos-neg 7.6%

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

    2. associate-*r* 7.6%

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

    3. *-commutative 7.6%

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

    4. *-commutative 7.6%

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

    5. sin-neg 7.6%

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

11. Simplified 7.6%

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

12. Taylor expanded in M around 0 8.2%

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

Reproduce

herbie shell --seed 2024191 
(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)))))))