Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.8%

Time: 15.6s

Alternatives: 11

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_1 := \left(m + n\right) \cdot 0.5\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot \cos \left({\left(\sqrt[3]{K \cdot t\_1}\right)}^{3} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + {\left(t\_1 - M\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0))))
        (t_1 (* (+ m n) 0.5)))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) INFINITY)
     (* t_0 (cos (- (pow (cbrt (* K t_1)) 3.0) M)))
     (exp (- (- m n) (+ l (pow (- t_1 M) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double t_1 = (m + n) * 0.5;
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= ((double) INFINITY)) {
		tmp = t_0 * cos((pow(cbrt((K * t_1)), 3.0) - M));
	} else {
		tmp = exp(((m - n) - (l + pow((t_1 - M), 2.0))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double t_1 = (m + n) * 0.5;
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.cos((Math.pow(Math.cbrt((K * t_1)), 3.0) - M));
	} else {
		tmp = Math.exp(((m - n) - (l + Math.pow((t_1 - M), 2.0))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	t_1 = Float64(Float64(m + n) * 0.5)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = Float64(t_0 * cos(Float64((cbrt(Float64(K * t_1)) ^ 3.0) - M)));
	else
		tmp = exp(Float64(Float64(m - n) - Float64(l + (Float64(t_1 - M) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(t$95$0 * N[Cos[N[(N[Power[N[Power[N[(K * t$95$1), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[Power[N[(t$95$1 - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 95.9%

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

      1. add-cube-cbrt 96.8%

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

      2. pow3 96.9%

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

      3. div-inv 96.9%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\left(K \cdot \left(m + n\right)\right) \cdot \frac{1}{2}}}\right)}^{3} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. associate-*r* 96.9%

         \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{K \cdot \left(\left(m + n\right) \cdot \frac{1}{2}\right)}}\right)}^{3} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. metadata-eval 96.9%

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

   4. Applied egg-rr 96.9%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0 98.0%

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

   4. Taylor expanded in M around 0 98.0%

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

   5. Taylor expanded in m around -inf 98.0%

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

      1. fabs-neg 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. fabs-sub 98.0%

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

      5. rem-square-sqrt 50.0%

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

      6. fabs-sqr 50.0%

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

      7. rem-square-sqrt 98.0%

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

   7. Simplified 98.0%

      \[\leadsto 1 \cdot e^{\color{blue}{\left(m - n\right)} - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left({\left(\sqrt[3]{K \cdot \left(\left(m + n\right) \cdot 0.5\right)}\right)}^{3} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{1}{\frac{\frac{2}{K}}{m + n}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0)))))
   (if (<= (* (cos (- (/ (* K (+ m n)) 2.0) M)) t_0) INFINITY)
     (* t_0 (cos (- (/ 1.0 (/ (/ 2.0 K) (+ m n))) M)))
     (exp (- (- m n) (+ l (pow (- (* (+ m n) 0.5) M) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= ((double) INFINITY)) {
		tmp = t_0 * cos(((1.0 / ((2.0 / K) / (m + n))) - M));
	} else {
		tmp = exp(((m - n) - (l + pow((((m + n) * 0.5) - M), 2.0))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if ((Math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.cos(((1.0 / ((2.0 / K) / (m + n))) - M));
	} else {
		tmp = Math.exp(((m - n) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if (math.cos((((K * (m + n)) / 2.0) - M)) * t_0) <= math.inf:
		tmp = t_0 * math.cos(((1.0 / ((2.0 / K) / (m + n))) - M))
	else:
		tmp = math.exp(((m - n) - (l + math.pow((((m + n) * 0.5) - M), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = Float64(t_0 * cos(Float64(Float64(1.0 / Float64(Float64(2.0 / K) / Float64(m + n))) - M)));
	else
		tmp = exp(Float64(Float64(m - n) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if ((cos((((K * (m + n)) / 2.0) - M)) * t_0) <= Inf)
		tmp = t_0 * cos(((1.0 / ((2.0 / K) / (m + n))) - M));
	else
		tmp = exp(((m - n) - (l + ((((m + n) * 0.5) - M) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(t$95$0 * N[Cos[N[(N[(1.0 / N[(N[(2.0 / K), $MachinePrecision] / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 95.9%

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

      1. clear-num 95.9%

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

      2. inv-pow 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. unpow-1 95.9%

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

      2. associate-/r* 96.3%

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

   6. Simplified 96.3%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0 98.0%

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

   4. Taylor expanded in M around 0 98.0%

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

   5. Taylor expanded in m around -inf 98.0%

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

      1. fabs-neg 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. fabs-sub 98.0%

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

      5. rem-square-sqrt 50.0%

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

      6. fabs-sqr 50.0%

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

      7. rem-square-sqrt 98.0%

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

   7. Simplified 98.0%

      \[\leadsto 1 \cdot e^{\color{blue}{\left(m - n\right)} - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\frac{1}{\frac{\frac{2}{K}}{m + n}} - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (fabs (- m n)) l) (pow (- (/ (+ m n) 2.0) M) 2.0))))))
   (if (<= t_0 INFINITY)
     t_0
     (exp (- (- m n) (+ l (pow (- (* (+ m n) 0.5) M) 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos((((K * (m + n)) / 2.0) - M)) * exp(((fabs((m - n)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = exp(((m - n) - (l + pow((((m + n) * 0.5) - M), 2.0))));
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = Math.exp(((m - n) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp(((math.fabs((m - n)) - l) - math.pow((((m + n) / 2.0) - M), 2.0)))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = math.exp(((m - n) - (l + math.pow((((m + n) * 0.5) - M), 2.0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = exp(Float64(Float64(m - n) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos((((K * (m + n)) / 2.0) - M)) * exp(((abs((m - n)) - l) - ((((m + n) / 2.0) - M) ^ 2.0)));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = exp(((m - n) - (l + ((((m + n) * 0.5) - M) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n)))))) < +inf.0

   1. Initial program 95.9%

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

   if +inf.0 < (*.f64 (cos.f64 (-.f64 (/.f64 (*.f64 K (+.f64 m n)) #s(literal 2 binary64)) M)) (exp.f64 (-.f64 (neg.f64 (pow.f64 (-.f64 (/.f64 (+.f64 m n) #s(literal 2 binary64)) M) #s(literal 2 binary64))) (-.f64 l (fabs.f64 (-.f64 m n))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0 98.0%

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

   4. Taylor expanded in M around 0 98.0%

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

   5. Taylor expanded in m around -inf 98.0%

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

      1. fabs-neg 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. fabs-sub 98.0%

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

      5. rem-square-sqrt 50.0%

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

      6. fabs-sqr 50.0%

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

      7. rem-square-sqrt 98.0%

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

   7. Simplified 98.0%

      \[\leadsto 1 \cdot e^{\color{blue}{\left(m - n\right)} - \left(\ell + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \leq \infty:\\ \;\;\;\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 1.4× speedup

Math

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

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(abs(Float64(m - n)) - Float64(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[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in K around 0 94.5%

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

4. Taylor expanded in M around 0 94.5%

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

5. Final simplification 94.5%

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

Alternative 5: 96.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (exp (- (- 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(((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(((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(((m - n) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in K around 0 94.5%

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

4. Taylor expanded in M around 0 94.5%

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

5. Taylor expanded in m around -inf 94.5%

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

   1. fabs-neg 94.5%

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

   2. mul-1-neg 94.5%

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

   3. sub-neg 94.5%

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

   4. fabs-sub 94.5%

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

   5. rem-square-sqrt 47.5%

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

   6. fabs-sqr 47.5%

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

   7. rem-square-sqrt 94.3%

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

7. Simplified 94.3%

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

8. Final simplification 94.3%

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

Alternative 6: 74.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in K around 0 94.5%

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

4. Taylor expanded in M around 0 94.5%

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

5. Taylor expanded in m around -inf 94.5%

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

   1. fabs-neg 94.5%

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

   2. mul-1-neg 94.5%

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

   3. sub-neg 94.5%

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

   4. fabs-sub 94.5%

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

   5. rem-square-sqrt 47.5%

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

   6. fabs-sqr 47.5%

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

   7. rem-square-sqrt 94.3%

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

7. Simplified 94.3%

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

8. Taylor expanded in M around inf 69.9%

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

9. Final simplification 69.9%

   \[\leadsto e^{\left(m - n\right) - \left({M}^{2} + \ell\right)} \]
10. Add Preprocessing

Alternative 7: 86.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((((m - n) - l) + ((-0.25d0) * ((m + n) * (m + n)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in K around 0 94.5%

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

4. Taylor expanded in M around 0 94.5%

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

5. Taylor expanded in M around 0 86.1%

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

   1. associate--r+ 86.1%

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

   2. cancel-sign-sub-inv 86.1%

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

   3. rem-square-sqrt 42.2%

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

   4. fabs-sqr 42.2%

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

   5. rem-square-sqrt 85.9%

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

   6. metadata-eval 85.9%

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

   7. +-commutative 85.9%

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

7. Simplified 85.9%

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

   1. +-commutative 85.9%

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

   2. pow2 85.9%

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

9. Applied egg-rr 85.9%

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

10. Final simplification 85.9%

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

Alternative 8: 6.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(0.5 \cdot \left(K \cdot n\right) - M\right) \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (cos (- (* 0.5 (* K n)) M)))

C

double code(double K, double m, double n, double M, double l) {
	return cos(((0.5 * (K * n)) - 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(((0.5d0 * (k * n)) - m_1))
end function

Java

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

Python

def code(K, m, n, M, l):
	return math.cos(((0.5 * (K * n)) - M))

Julia

function code(K, m, n, M, l)
	return cos(Float64(Float64(0.5 * Float64(K * n)) - M))
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[N[(N[(0.5 * N[(K * n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in l around inf 28.5%

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

   1. mul-1-neg 28.5%

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

5. Simplified 28.5%

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

6. Taylor expanded in l around 0 7.5%

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

7. Taylor expanded in m around 0 7.6%

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

   1. *-commutative 7.6%

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

9. Simplified 7.6%

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

10. Final simplification 7.6%

    \[\leadsto \cos \left(0.5 \cdot \left(K \cdot n\right) - M\right) \]
11. Add Preprocessing

Alternative 9: 6.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(0.5 \cdot \left(K \cdot n\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (cos (* 0.5 (* K n))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((0.5 * (K * 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((0.5d0 * (k * n)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((0.5 * (K * n)));
}

Python

def code(K, m, n, M, l):
	return math.cos((0.5 * (K * n)))

Julia

function code(K, m, n, M, l)
	return cos(Float64(0.5 * Float64(K * n)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((0.5 * (K * n)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[N[(0.5 * N[(K * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in l around inf 28.5%

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

   1. mul-1-neg 28.5%

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

5. Simplified 28.5%

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

6. Taylor expanded in l around 0 7.5%

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

   1. +-commutative 7.5%

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

   2. add-cbrt-cube 5.9%

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot \color{blue}{\sqrt[3]{\left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(n + m\right)}}\right) - M\right) \]

   3. pow1/3 4.3%

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot \color{blue}{{\left(\left(\left(n + m\right) \cdot \left(n + m\right)\right) \cdot \left(n + m\right)\right)}^{0.3333333333333333}}\right) - M\right) \]

   4. +-commutative 4.3%

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot {\left(\left(\color{blue}{\left(m + n\right)} \cdot \left(n + m\right)\right) \cdot \left(n + m\right)\right)}^{0.3333333333333333}\right) - M\right) \]

   5. +-commutative 4.3%

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot {\left(\left(\left(m + n\right) \cdot \color{blue}{\left(m + n\right)}\right) \cdot \left(n + m\right)\right)}^{0.3333333333333333}\right) - M\right) \]

   6. +-commutative 4.3%

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot {\left(\left(\left(m + n\right) \cdot \left(m + n\right)\right) \cdot \color{blue}{\left(m + n\right)}\right)}^{0.3333333333333333}\right) - M\right) \]

   7. pow3 4.3%

      \[\leadsto \cos \left(0.5 \cdot \left(K \cdot {\color{blue}{\left({\left(m + n\right)}^{3}\right)}}^{0.3333333333333333}\right) - M\right) \]

8. Applied egg-rr 4.3%

   \[\leadsto \cos \left(0.5 \cdot \left(K \cdot \color{blue}{{\left({\left(m + n\right)}^{3}\right)}^{0.3333333333333333}}\right) - M\right) \]

9. Taylor expanded in n around inf 7.6%

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

    1. *-commutative 7.6%

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

11. Simplified 7.6%

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

12. Final simplification 7.6%

    \[\leadsto \cos \left(0.5 \cdot \left(K \cdot n\right)\right) \]
13. Add Preprocessing

Alternative 10: 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 77.2%

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

3. Taylor expanded in l around inf 28.5%

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

   1. mul-1-neg 28.5%

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

5. Simplified 28.5%

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

6. Taylor expanded in l around 0 7.5%

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

7. Taylor expanded in K around 0 7.4%

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

   1. cos-neg 7.4%

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

9. Simplified 7.4%

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

Alternative 11: 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 77.2%

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

3. Taylor expanded in l around inf 28.5%

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

   1. mul-1-neg 28.5%

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

5. Simplified 28.5%

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

6. Taylor expanded in l around 0 7.5%

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

7. Taylor expanded in K around 0 7.4%

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

   1. cos-neg 7.4%

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

9. Simplified 7.4%

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

10. Taylor expanded in M around 0 7.4%

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

Reproduce

herbie shell --seed 2024137 
(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)))))))