Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.7% → 96.8%

Time: 14.2s

Alternatives: 6

Speedup: 3.6×

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, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = (m + n) * 0.5d0
    code = cos(m_1) * exp((((t_0 - m_1) * (m_1 - t_0)) + (abs((m - n)) - 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;
	return Math.cos(M) * Math.exp((((t_0 - M) * (M - t_0)) + (Math.abs((m - n)) - l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in K around 0 98.9%

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

   1. cos-neg 98.9%

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

5. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-commutative 98.9%

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

   4. unpow2 98.9%

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

   5. *-commutative 98.9%

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

   6. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 96.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = (m + n) * 0.5d0
    code = exp((((t_0 - m_1) * (m_1 - t_0)) + (abs((m - n)) - 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;
	return Math.exp((((t_0 - M) * (M - t_0)) + (Math.abs((m - n)) - l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in K around 0 98.9%

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

   1. cos-neg 98.9%

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

5. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-commutative 98.9%

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

   4. unpow2 98.9%

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

   5. *-commutative 98.9%

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

   6. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in M around 0 97.7%

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

9. Final simplification 97.7%

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

Alternative 3: 96.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * 0.5;
	return exp((((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 = (m + n) * 0.5d0
    code = exp((((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 = (m + n) * 0.5;
	return Math.exp((((t_0 - M) * (M - t_0)) - l));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in K around 0 98.9%

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

   1. cos-neg 98.9%

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

5. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-commutative 98.9%

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

   4. unpow2 98.9%

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

   5. *-commutative 98.9%

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

   6. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in M around 0 97.7%

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

9. Taylor expanded in l around inf 97.4%

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

10. Final simplification 97.4%

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

Alternative 4: 35.6% 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 75.5%

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

3. Taylor expanded in l around inf 28.4%

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

   1. mul-1-neg 28.4%

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

5. Simplified 28.4%

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

6. Taylor expanded in n around inf 26.3%

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

7. Taylor expanded in K around inf 26.2%

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

8. Taylor expanded in K around 0 32.7%

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

Alternative 5: 7.3% 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 75.5%

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

3. Taylor expanded in l around inf 28.4%

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

   1. mul-1-neg 28.4%

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

5. Simplified 28.4%

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

6. Taylor expanded in l around 0 4.8%

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

   1. associate-*r* 4.8%

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

8. Simplified 4.8%

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

9. Taylor expanded in K around 0 5.5%

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

    1. cos-neg 5.5%

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

11. Simplified 5.5%

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

Alternative 6: 7.3% 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 75.5%

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

3. Taylor expanded in l around inf 28.4%

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

   1. mul-1-neg 28.4%

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

5. Simplified 28.4%

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

6. Taylor expanded in l around 0 4.8%

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

   1. associate-*r* 4.8%

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

8. Simplified 4.8%

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

9. Taylor expanded in K around 0 5.5%

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

    1. cos-neg 5.5%

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

11. Simplified 5.5%

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

12. Taylor expanded in M around 0 5.5%

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

Reproduce

herbie shell --seed 2024182 
(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)))))))