Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.3%

Time: 7.3s

Alternatives: 9

Speedup: 2.6×

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: 75.9% 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: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * exp((fabs((n - m)) - (pow(fma(0.5, (n + m), -M), 2.0) + l)));
}

Julia

function code(K, m, n, M, l)
	return Float64(1.0 * exp(Float64(abs(Float64(n - m)) - Float64((fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0) + l))))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 97.8%

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

9. Final simplification 97.8%

   \[\leadsto 1 \cdot e^{\left|n - m\right| - \left({\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2} + \ell\right)} \]
10. Add Preprocessing

Alternative 2: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := e^{t\_0 - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)}\\ \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+148}:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;e^{t\_0 - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))) (t_1 (exp (- t_0 (+ (* 0.25 (* n n)) l)))))
   (if (<= l -8.8e+148)
     (* t_1 1.0)
     (if (<= l 2.3e+68)
       (* (exp (- t_0 (pow (fma 0.5 (+ n m) (- M)) 2.0))) 1.0)
       (* (cos M) t_1)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = exp((t_0 - ((0.25 * (n * n)) + l)));
	double tmp;
	if (l <= -8.8e+148) {
		tmp = t_1 * 1.0;
	} else if (l <= 2.3e+68) {
		tmp = exp((t_0 - pow(fma(0.5, (n + m), -M), 2.0))) * 1.0;
	} else {
		tmp = cos(M) * t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = exp(Float64(t_0 - Float64(Float64(0.25 * Float64(n * n)) + l)))
	tmp = 0.0
	if (l <= -8.8e+148)
		tmp = Float64(t_1 * 1.0);
	elseif (l <= 2.3e+68)
		tmp = Float64(exp(Float64(t_0 - (fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0))) * 1.0);
	else
		tmp = Float64(cos(M) * t_1);
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(t$95$0 - N[(N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -8.8e+148], N[(t$95$1 * 1.0), $MachinePrecision], If[LessEqual[l, 2.3e+68], N[(N[Exp[N[(t$95$0 - N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.7999999999999995e148

   1. Initial program 69.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\left|m - n\right| - \left(\frac{1}{4} \cdot {n}^{2} + \ell\right)} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 76.1%

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

   if -8.7999999999999995e148 < l < 2.3e68

   1. Initial program 73.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in l around 0

      \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 95.4%

       \[\leadsto e^{\left|m - n\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1 \]

   if 2.3e68 < l

   1. Initial program 92.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 97.6%

      \[\leadsto e^{\left|m - n\right| - \left(\left(n \cdot n\right) \cdot 0.25 + \ell\right)} \cdot \cos M \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+148}:\\ \;\;\;\;e^{\left|n - m\right| - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)} \cdot 1\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;e^{\left|n - m\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := e^{t\_0 - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)} \cdot 1\\ \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;e^{t\_0 - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m)))
        (t_1 (* (exp (- t_0 (+ (* 0.25 (* n n)) l))) 1.0)))
   (if (<= l -8.8e+148)
     t_1
     (if (<= l 2.3e+68)
       (* (exp (- t_0 (pow (fma 0.5 (+ n m) (- M)) 2.0))) 1.0)
       t_1))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = exp((t_0 - ((0.25 * (n * n)) + l))) * 1.0;
	double tmp;
	if (l <= -8.8e+148) {
		tmp = t_1;
	} else if (l <= 2.3e+68) {
		tmp = exp((t_0 - pow(fma(0.5, (n + m), -M), 2.0))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(exp(Float64(t_0 - Float64(Float64(0.25 * Float64(n * n)) + l))) * 1.0)
	tmp = 0.0
	if (l <= -8.8e+148)
		tmp = t_1;
	elseif (l <= 2.3e+68)
		tmp = Float64(exp(Float64(t_0 - (fma(0.5, Float64(n + m), Float64(-M)) ^ 2.0))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(t$95$0 - N[(N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[l, -8.8e+148], t$95$1, If[LessEqual[l, 2.3e+68], N[(N[Exp[N[(t$95$0 - N[Power[N[(0.5 * N[(n + m), $MachinePrecision] + (-M)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if l < -8.7999999999999995e148 or 2.3e68 < l

   1. Initial program 82.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\left|m - n\right| - \left(\frac{1}{4} \cdot {n}^{2} + \ell\right)} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 88.0%

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

   if -8.7999999999999995e148 < l < 2.3e68

   1. Initial program 73.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in l around 0

      \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 95.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+148}:\\ \;\;\;\;e^{\left|n - m\right| - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)} \cdot 1\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;e^{\left|n - m\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 5.3 \cdot 10^{-61}:\\ \;\;\;\;e^{t\_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot 1\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+28}:\\ \;\;\;\;e^{t\_0 - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 5.3e-61)
     (* (exp (- t_0 (+ (* (* m m) 0.25) l))) 1.0)
     (if (<= n 4e+28)
       (* (exp (- t_0 (+ (* M M) l))) 1.0)
       (* (exp (- t_0 (+ (* 0.25 (* n n)) l))) 1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 5.3e-61) {
		tmp = exp((t_0 - (((m * m) * 0.25) + l))) * 1.0;
	} else if (n <= 4e+28) {
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = exp((t_0 - ((0.25 * (n * n)) + l))) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 5.3d-61) then
        tmp = exp((t_0 - (((m * m) * 0.25d0) + l))) * 1.0d0
    else if (n <= 4d+28) then
        tmp = exp((t_0 - ((m_1 * m_1) + l))) * 1.0d0
    else
        tmp = exp((t_0 - ((0.25d0 * (n * n)) + l))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 5.3e-61) {
		tmp = Math.exp((t_0 - (((m * m) * 0.25) + l))) * 1.0;
	} else if (n <= 4e+28) {
		tmp = Math.exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = Math.exp((t_0 - ((0.25 * (n * n)) + l))) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 5.3e-61:
		tmp = math.exp((t_0 - (((m * m) * 0.25) + l))) * 1.0
	elif n <= 4e+28:
		tmp = math.exp((t_0 - ((M * M) + l))) * 1.0
	else:
		tmp = math.exp((t_0 - ((0.25 * (n * n)) + l))) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 5.3e-61)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))) * 1.0);
	elseif (n <= 4e+28)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(M * M) + l))) * 1.0);
	else
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(0.25 * Float64(n * n)) + l))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 5.3e-61)
		tmp = exp((t_0 - (((m * m) * 0.25) + l))) * 1.0;
	elseif (n <= 4e+28)
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	else
		tmp = exp((t_0 - ((0.25 * (n * n)) + l))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 5.3e-61], N[(N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[n, 4e+28], N[(N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(t$95$0 - N[(N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.3e-61

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\left|m - n\right| - \left(\frac{1}{4} \cdot {m}^{2} + \ell\right)} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 67.2%

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

   if 5.3e-61 < n < 3.99999999999999983e28

   1. Initial program 76.6%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 95.3%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 60.5%

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

   if 3.99999999999999983e28 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in n around inf

      \[\leadsto e^{\left|m - n\right| - \left(\frac{1}{4} \cdot {n}^{2} + \ell\right)} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.3 \cdot 10^{-61}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot 1\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+28}:\\ \;\;\;\;e^{\left|n - m\right| - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 5.3 \cdot 10^{-61}:\\ \;\;\;\;e^{t\_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot 1\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{+52}:\\ \;\;\;\;e^{t\_0 - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - 0.25 \cdot \left(n \cdot n\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 5.3e-61)
     (* (exp (- t_0 (+ (* (* m m) 0.25) l))) 1.0)
     (if (<= n 8.2e+52)
       (* (exp (- t_0 (+ (* M M) l))) 1.0)
       (* (exp (- t_0 (* 0.25 (* n n)))) 1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 5.3e-61) {
		tmp = exp((t_0 - (((m * m) * 0.25) + l))) * 1.0;
	} else if (n <= 8.2e+52) {
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = exp((t_0 - (0.25 * (n * n)))) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 5.3d-61) then
        tmp = exp((t_0 - (((m * m) * 0.25d0) + l))) * 1.0d0
    else if (n <= 8.2d+52) then
        tmp = exp((t_0 - ((m_1 * m_1) + l))) * 1.0d0
    else
        tmp = exp((t_0 - (0.25d0 * (n * n)))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 5.3e-61) {
		tmp = Math.exp((t_0 - (((m * m) * 0.25) + l))) * 1.0;
	} else if (n <= 8.2e+52) {
		tmp = Math.exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = Math.exp((t_0 - (0.25 * (n * n)))) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 5.3e-61:
		tmp = math.exp((t_0 - (((m * m) * 0.25) + l))) * 1.0
	elif n <= 8.2e+52:
		tmp = math.exp((t_0 - ((M * M) + l))) * 1.0
	else:
		tmp = math.exp((t_0 - (0.25 * (n * n)))) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 5.3e-61)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))) * 1.0);
	elseif (n <= 8.2e+52)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(M * M) + l))) * 1.0);
	else
		tmp = Float64(exp(Float64(t_0 - Float64(0.25 * Float64(n * n)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 5.3e-61)
		tmp = exp((t_0 - (((m * m) * 0.25) + l))) * 1.0;
	elseif (n <= 8.2e+52)
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	else
		tmp = exp((t_0 - (0.25 * (n * n)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 5.3e-61], N[(N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[n, 8.2e+52], N[(N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(t$95$0 - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.3e-61

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in m around inf

      \[\leadsto e^{\left|m - n\right| - \left(\frac{1}{4} \cdot {m}^{2} + \ell\right)} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 67.2%

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

   if 5.3e-61 < n < 8.1999999999999999e52

   1. Initial program 76.1%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 69.2%

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

   if 8.1999999999999999e52 < n

   1. Initial program 65.6%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in l around 0

      \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto e^{\left|m - n\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1 \]

   12. Taylor expanded in n around inf

       \[\leadsto e^{\left|m - n\right| - \frac{1}{4} \cdot {n}^{2}} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 96.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.3 \cdot 10^{-61}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)} \cdot 1\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{+52}:\\ \;\;\;\;e^{\left|n - m\right| - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 4.2 \cdot 10^{-295}:\\ \;\;\;\;e^{t\_0 - \left(m \cdot m\right) \cdot 0.25} \cdot 1\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{+52}:\\ \;\;\;\;e^{t\_0 - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - 0.25 \cdot \left(n \cdot n\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 4.2e-295)
     (* (exp (- t_0 (* (* m m) 0.25))) 1.0)
     (if (<= n 8.2e+52)
       (* (exp (- t_0 (+ (* M M) l))) 1.0)
       (* (exp (- t_0 (* 0.25 (* n n)))) 1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (n <= 4.2e-295) {
		tmp = exp((t_0 - ((m * m) * 0.25))) * 1.0;
	} else if (n <= 8.2e+52) {
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = exp((t_0 - (0.25 * (n * n)))) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (n <= 4.2d-295) then
        tmp = exp((t_0 - ((m * m) * 0.25d0))) * 1.0d0
    else if (n <= 8.2d+52) then
        tmp = exp((t_0 - ((m_1 * m_1) + l))) * 1.0d0
    else
        tmp = exp((t_0 - (0.25d0 * (n * n)))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 4.2e-295) {
		tmp = Math.exp((t_0 - ((m * m) * 0.25))) * 1.0;
	} else if (n <= 8.2e+52) {
		tmp = Math.exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = Math.exp((t_0 - (0.25 * (n * n)))) * 1.0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 4.2e-295:
		tmp = math.exp((t_0 - ((m * m) * 0.25))) * 1.0
	elif n <= 8.2e+52:
		tmp = math.exp((t_0 - ((M * M) + l))) * 1.0
	else:
		tmp = math.exp((t_0 - (0.25 * (n * n)))) * 1.0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 4.2e-295)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(m * m) * 0.25))) * 1.0);
	elseif (n <= 8.2e+52)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(M * M) + l))) * 1.0);
	else
		tmp = Float64(exp(Float64(t_0 - Float64(0.25 * Float64(n * n)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (n <= 4.2e-295)
		tmp = exp((t_0 - ((m * m) * 0.25))) * 1.0;
	elseif (n <= 8.2e+52)
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	else
		tmp = exp((t_0 - (0.25 * (n * n)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 4.2e-295], N[(N[Exp[N[(t$95$0 - N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[n, 8.2e+52], N[(N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(t$95$0 - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 4.19999999999999986e-295

   1. Initial program 76.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.2%

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

   9. Taylor expanded in l around 0

      \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 83.6%

       \[\leadsto e^{\left|m - n\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1 \]

   12. Taylor expanded in m around inf

       \[\leadsto e^{\left|m - n\right| - \frac{1}{4} \cdot {m}^{2}} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 55.2%

       \[\leadsto e^{\left|m - n\right| - \left(m \cdot m\right) \cdot 0.25} \cdot 1 \]

   if 4.19999999999999986e-295 < n < 8.1999999999999999e52

   1. Initial program 86.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 76.8%

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

   if 8.1999999999999999e52 < n

   1. Initial program 65.6%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in l around 0

      \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto e^{\left|m - n\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1 \]

   12. Taylor expanded in n around inf

       \[\leadsto e^{\left|m - n\right| - \frac{1}{4} \cdot {n}^{2}} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 96.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-295}:\\ \;\;\;\;e^{\left|n - m\right| - \left(m \cdot m\right) \cdot 0.25} \cdot 1\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{+52}:\\ \;\;\;\;e^{\left|n - m\right| - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := e^{t\_0 - \left(m \cdot m\right) \cdot 0.25} \cdot 1\\ \mathbf{if}\;m \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;m \leq 4 \cdot 10^{+81}:\\ \;\;\;\;e^{t\_0 - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))) (t_1 (* (exp (- t_0 (* (* m m) 0.25))) 1.0)))
   (if (<= m -5e+39)
     t_1
     (if (<= m 4e+81) (* (exp (- t_0 (+ (* M M) l))) 1.0) t_1))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = exp((t_0 - ((m * m) * 0.25))) * 1.0;
	double tmp;
	if (m <= -5e+39) {
		tmp = t_1;
	} else if (m <= 4e+81) {
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((n - m))
    t_1 = exp((t_0 - ((m * m) * 0.25d0))) * 1.0d0
    if (m <= (-5d+39)) then
        tmp = t_1
    else if (m <= 4d+81) then
        tmp = exp((t_0 - ((m_1 * m_1) + l))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double t_1 = Math.exp((t_0 - ((m * m) * 0.25))) * 1.0;
	double tmp;
	if (m <= -5e+39) {
		tmp = t_1;
	} else if (m <= 4e+81) {
		tmp = Math.exp((t_0 - ((M * M) + l))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	t_1 = math.exp((t_0 - ((m * m) * 0.25))) * 1.0
	tmp = 0
	if m <= -5e+39:
		tmp = t_1
	elif m <= 4e+81:
		tmp = math.exp((t_0 - ((M * M) + l))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(exp(Float64(t_0 - Float64(Float64(m * m) * 0.25))) * 1.0)
	tmp = 0.0
	if (m <= -5e+39)
		tmp = t_1;
	elseif (m <= 4e+81)
		tmp = Float64(exp(Float64(t_0 - Float64(Float64(M * M) + l))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	t_1 = exp((t_0 - ((m * m) * 0.25))) * 1.0;
	tmp = 0.0;
	if (m <= -5e+39)
		tmp = t_1;
	elseif (m <= 4e+81)
		tmp = exp((t_0 - ((M * M) + l))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(t$95$0 - N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[m, -5e+39], t$95$1, If[LessEqual[m, 4e+81], N[(N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.00000000000000015e39 or 3.99999999999999969e81 < m

   1. Initial program 73.4%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in l around 0

      \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto e^{\left|m - n\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1 \]

   12. Taylor expanded in m around inf

       \[\leadsto e^{\left|m - n\right| - \frac{1}{4} \cdot {m}^{2}} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 91.6%

       \[\leadsto e^{\left|m - n\right| - \left(m \cdot m\right) \cdot 0.25} \cdot 1 \]

   if -5.00000000000000015e39 < m < 3.99999999999999969e81

   1. Initial program 77.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 66.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+39}:\\ \;\;\;\;e^{\left|n - m\right| - \left(m \cdot m\right) \cdot 0.25} \cdot 1\\ \mathbf{elif}\;m \leq 4 \cdot 10^{+81}:\\ \;\;\;\;e^{\left|n - m\right| - \left(M \cdot M + \ell\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(m \cdot m\right) \cdot 0.25} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - \left(M \cdot M + \ell\right)} \cdot 1 \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (exp (- (fabs (- n m)) (+ (* M M) l))) 1.0))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - ((M * M) + l))) * 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 = exp((abs((n - m)) - ((m_1 * m_1) + l))) * 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.abs((n - m)) - ((M * M) + l))) * 1.0;
}

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) - ((M * M) + l))) * 1.0

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(abs(Float64(n - m)) - Float64(Float64(M * M) + l))) * 1.0)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((abs((n - m)) - ((M * M) + l))) * 1.0;
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 97.8%

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

9. Taylor expanded in M around inf

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

11. Applied rewrites 55.5%

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

12. Final simplification 55.5%

    \[\leadsto e^{\left|n - m\right| - \left(M \cdot M + \ell\right)} \cdot 1 \]
13. Add Preprocessing

Alternative 9: 44.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ e^{\left|n - m\right| - M \cdot M} \cdot 1 \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (exp (- (fabs (- n m)) (* M M))) 1.0))

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (M * M))) * 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 = exp((abs((n - m)) - (m_1 * m_1))) * 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((Math.abs((n - m)) - (M * M))) * 1.0;
}

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) - (M * M))) * 1.0

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(abs(Float64(n - m)) - Float64(M * M))) * 1.0)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((abs((n - m)) - (M * M))) * 1.0;
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 97.8%

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

9. Taylor expanded in l around 0

   \[\leadsto e^{\left|m - n\right| - {\left(\frac{1}{2} \cdot \left(m + n\right) - M\right)}^{2}} \cdot 1 \]
10. Step-by-step derivation

11. Applied rewrites 86.8%

    \[\leadsto e^{\left|m - n\right| - {\left(\mathsf{fma}\left(0.5, n + m, -M\right)\right)}^{2}} \cdot 1 \]

12. Taylor expanded in M around inf

    \[\leadsto e^{\left|m - n\right| - {M}^{2}} \cdot 1 \]
13. Step-by-step derivation

14. Applied rewrites 43.7%

    \[\leadsto e^{\left|m - n\right| - M \cdot M} \cdot 1 \]

15. Final simplification 43.7%

    \[\leadsto e^{\left|n - m\right| - M \cdot M} \cdot 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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)))))))