Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.6%

Time: 13.5s

Alternatives: 9

Speedup: 2.7×

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: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos 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(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(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(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(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(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(M) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $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]

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 94.7

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

5. Applied rewrites 94.7%

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

Alternative 2: 92.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ t_1 := \left|m - n\right|\\ t_2 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_1\right)}\\ t_3 := 1 \cdot \left(\left(\left(-\ell\right) + 1\right) \cdot e^{t\_1 - t\_0 \cdot t\_0}\right)\\ \mathbf{if}\;t\_2 \leq 0.8:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M))
        (t_1 (fabs (- m n)))
        (t_2
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_1)))))
        (t_3 (* 1.0 (* (+ (- l) 1.0) (exp (- t_1 (* t_0 t_0)))))))
   (if (<= t_2 0.8)
     t_3
     (if (<= t_2 INFINITY) (* (cos (* 0.5 (* (+ n m) K))) (exp (- l))) t_3))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = fabs((m - n));
	double t_2 = cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double t_3 = 1.0 * ((-l + 1.0) * exp((t_1 - (t_0 * t_0))));
	double tmp;
	if (t_2 <= 0.8) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = cos((0.5 * ((n + m) * K))) * exp(-l);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = Math.abs((m - n));
	double t_2 = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double t_3 = 1.0 * ((-l + 1.0) * Math.exp((t_1 - (t_0 * t_0))));
	double tmp;
	if (t_2 <= 0.8) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.cos((0.5 * ((n + m) * K))) * Math.exp(-l);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (0.5 * (n + m)) - M
	t_1 = math.fabs((m - n))
	t_2 = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)))
	t_3 = 1.0 * ((-l + 1.0) * math.exp((t_1 - (t_0 * t_0))))
	tmp = 0
	if t_2 <= 0.8:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = math.cos((0.5 * ((n + m) * K))) * math.exp(-l)
	else:
		tmp = t_3
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	t_1 = abs(Float64(m - n))
	t_2 = 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 - t_1))))
	t_3 = Float64(1.0 * Float64(Float64(Float64(-l) + 1.0) * exp(Float64(t_1 - Float64(t_0 * t_0)))))
	tmp = 0.0
	if (t_2 <= 0.8)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(cos(Float64(0.5 * Float64(Float64(n + m) * K))) * exp(Float64(-l)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (0.5 * (n + m)) - M;
	t_1 = abs((m - n));
	t_2 = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - t_1)));
	t_3 = 1.0 * ((-l + 1.0) * exp((t_1 - (t_0 * t_0))));
	tmp = 0.0;
	if (t_2 <= 0.8)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = cos((0.5 * ((n + m) * K))) * exp(-l);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * N[(N[((-l) + 1.0), $MachinePrecision] * N[Exp[N[(t$95$1 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.8], t$95$3, If[LessEqual[t$95$2, Infinity], N[(N[Cos[N[(0.5 * N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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)))))) < 0.80000000000000004 or +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 74.5%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 27.6

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 27.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-exp.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-fabs.f64 N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower--.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. lower-+.f64 N/A

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

   14. Applied rewrites 92.2%

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

   if 0.80000000000000004 < (*.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 86.3%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in K around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 86.3

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

   8. Applied rewrites 86.3%

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

Alternative 3: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ t_1 := \left|m - n\right|\\ t_2 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_1\right)}\\ t_3 := 1 \cdot \left(\left(\left(-\ell\right) + 1\right) \cdot e^{t\_1 - t\_0 \cdot t\_0}\right)\\ \mathbf{if}\;t\_2 \leq 0.9:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\cos \left(0.5 \cdot \left(n \cdot K\right)\right) \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M))
        (t_1 (fabs (- m n)))
        (t_2
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_1)))))
        (t_3 (* 1.0 (* (+ (- l) 1.0) (exp (- t_1 (* t_0 t_0)))))))
   (if (<= t_2 0.9)
     t_3
     (if (<= t_2 INFINITY) (* (cos (* 0.5 (* n K))) (exp (- l))) t_3))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = fabs((m - n));
	double t_2 = cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double t_3 = 1.0 * ((-l + 1.0) * exp((t_1 - (t_0 * t_0))));
	double tmp;
	if (t_2 <= 0.9) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = cos((0.5 * (n * K))) * exp(-l);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = Math.abs((m - n));
	double t_2 = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double t_3 = 1.0 * ((-l + 1.0) * Math.exp((t_1 - (t_0 * t_0))));
	double tmp;
	if (t_2 <= 0.9) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.cos((0.5 * (n * K))) * Math.exp(-l);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (0.5 * (n + m)) - M
	t_1 = math.fabs((m - n))
	t_2 = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)))
	t_3 = 1.0 * ((-l + 1.0) * math.exp((t_1 - (t_0 * t_0))))
	tmp = 0
	if t_2 <= 0.9:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = math.cos((0.5 * (n * K))) * math.exp(-l)
	else:
		tmp = t_3
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	t_1 = abs(Float64(m - n))
	t_2 = 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 - t_1))))
	t_3 = Float64(1.0 * Float64(Float64(Float64(-l) + 1.0) * exp(Float64(t_1 - Float64(t_0 * t_0)))))
	tmp = 0.0
	if (t_2 <= 0.9)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(cos(Float64(0.5 * Float64(n * K))) * exp(Float64(-l)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (0.5 * (n + m)) - M;
	t_1 = abs((m - n));
	t_2 = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - t_1)));
	t_3 = 1.0 * ((-l + 1.0) * exp((t_1 - (t_0 * t_0))));
	tmp = 0.0;
	if (t_2 <= 0.9)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = cos((0.5 * (n * K))) * exp(-l);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * N[(N[((-l) + 1.0), $MachinePrecision] * N[Exp[N[(t$95$1 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.9], t$95$3, If[LessEqual[t$95$2, Infinity], N[(N[Cos[N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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)))))) < 0.900000000000000022 or +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 74.1%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 27.5

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 27.5%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.0%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-exp.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-fabs.f64 N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower--.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. lower-+.f64 N/A

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

   14. Applied rewrites 91.9%

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

   if 0.900000000000000022 < (*.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 89.3%

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

   3. Taylor expanded in l around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \cos \color{blue}{\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right)} \cdot e^{\mathsf{neg}\left(\ell\right)} \]

      2. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\left(n \cdot K\right)}\right) \cdot e^{\mathsf{neg}\left(\ell\right)} \]

      3. lower-*.f64 86.9

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

   8. Applied rewrites 86.9%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ t_1 := \left|m - n\right|\\ t_2 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_1\right)}\\ t_3 := 1 \cdot \left(\left(\left(-\ell\right) + 1\right) \cdot e^{t\_1 - t\_0 \cdot t\_0}\right)\\ \mathbf{if}\;t\_2 \leq 0.9:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M))
        (t_1 (fabs (- m n)))
        (t_2
         (*
          (cos (- (/ (* K (+ m n)) 2.0) M))
          (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_1)))))
        (t_3 (* 1.0 (* (+ (- l) 1.0) (exp (- t_1 (* t_0 t_0)))))))
   (if (<= t_2 0.9) t_3 (if (<= t_2 INFINITY) (* 1.0 (exp (- l))) t_3))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = fabs((m - n));
	double t_2 = cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double t_3 = 1.0 * ((-l + 1.0) * exp((t_1 - (t_0 * t_0))));
	double tmp;
	if (t_2 <= 0.9) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 1.0 * exp(-l);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = Math.abs((m - n));
	double t_2 = Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)));
	double t_3 = 1.0 * ((-l + 1.0) * Math.exp((t_1 - (t_0 * t_0))));
	double tmp;
	if (t_2 <= 0.9) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (0.5 * (n + m)) - M
	t_1 = math.fabs((m - n))
	t_2 = math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - t_1)))
	t_3 = 1.0 * ((-l + 1.0) * math.exp((t_1 - (t_0 * t_0))))
	tmp = 0
	if t_2 <= 0.9:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_3
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	t_1 = abs(Float64(m - n))
	t_2 = 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 - t_1))))
	t_3 = Float64(1.0 * Float64(Float64(Float64(-l) + 1.0) * exp(Float64(t_1 - Float64(t_0 * t_0)))))
	tmp = 0.0
	if (t_2 <= 0.9)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (0.5 * (n + m)) - M;
	t_1 = abs((m - n));
	t_2 = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - t_1)));
	t_3 = 1.0 * ((-l + 1.0) * exp((t_1 - (t_0 * t_0))));
	tmp = 0.0;
	if (t_2 <= 0.9)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * N[(N[((-l) + 1.0), $MachinePrecision] * N[Exp[N[(t$95$1 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.9], t$95$3, If[LessEqual[t$95$2, Infinity], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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)))))) < 0.900000000000000022 or +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 74.1%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 27.5

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 27.5%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.0%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-exp.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-fabs.f64 N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower--.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. lower-+.f64 N/A

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

   14. Applied rewrites 91.9%

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

   if 0.900000000000000022 < (*.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 89.3%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 86.9

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 86.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.9%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right) - M\\ t_1 := \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)\\ t_2 := \left|m - n\right|\\ \mathbf{if}\;t\_1 \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - t\_2\right)} \leq \infty:\\ \;\;\;\;t\_1 \cdot e^{M \cdot \mathsf{fma}\left(M, \frac{\frac{\left|n - m\right| - \mathsf{fma}\left(0.25 \cdot \left(n + m\right), n + m, \ell\right)}{M} + \left(n + m\right)}{M}, -M\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\left(-\ell\right) + 1\right) \cdot e^{t\_2 - t\_0 \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ n m)) M))
        (t_1 (cos (- (/ (* K (+ m n)) 2.0) M)))
        (t_2 (fabs (- m n))))
   (if (<=
        (* t_1 (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l t_2))))
        INFINITY)
     (*
      t_1
      (exp
       (*
        M
        (fma
         M
         (/
          (+ (/ (- (fabs (- n m)) (fma (* 0.25 (+ n m)) (+ n m) l)) M) (+ n m))
          M)
         (- M)))))
     (* 1.0 (* (+ (- l) 1.0) (exp (- t_2 (* t_0 t_0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (n + m)) - M;
	double t_1 = cos((((K * (m + n)) / 2.0) - M));
	double t_2 = fabs((m - n));
	double tmp;
	if ((t_1 * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - t_2)))) <= ((double) INFINITY)) {
		tmp = t_1 * exp((M * fma(M, ((((fabs((n - m)) - fma((0.25 * (n + m)), (n + m), l)) / M) + (n + m)) / M), -M)));
	} else {
		tmp = 1.0 * ((-l + 1.0) * exp((t_2 - (t_0 * t_0))));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(n + m)) - M)
	t_1 = cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M))
	t_2 = abs(Float64(m - n))
	tmp = 0.0
	if (Float64(t_1 * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - t_2)))) <= Inf)
		tmp = Float64(t_1 * exp(Float64(M * fma(M, Float64(Float64(Float64(Float64(abs(Float64(n - m)) - fma(Float64(0.25 * Float64(n + m)), Float64(n + m), l)) / M) + Float64(n + m)) / M), Float64(-M)))));
	else
		tmp = Float64(1.0 * Float64(Float64(Float64(-l) + 1.0) * exp(Float64(t_2 - Float64(t_0 * t_0)))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[Exp[N[(M * N[(M * N[(N[(N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(N[(0.25 * N[(n + m), $MachinePrecision]), $MachinePrecision] * N[(n + m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision] + N[(n + m), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision] + (-M)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[((-l) + 1.0), $MachinePrecision] * N[Exp[N[(t$95$2 - N[(t$95$0 * t$95$0), $MachinePrecision]), $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 94.2%

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

   3. Taylor expanded in M around -inf

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

   5. Applied rewrites 92.7%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{M \cdot \mathsf{fma}\left(M, \frac{\frac{\left|n - m\right| - \mathsf{fma}\left(0.25 \cdot \left(n + m\right), n + m, \ell\right)}{M} + \left(n + m\right)}{M}, -M\right)}} \]

   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

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 33.9

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 33.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.9%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-exp.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-fabs.f64 N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower--.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. lower-+.f64 N/A

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

   14. Applied rewrites 100.0%

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

Alternative 6: 76.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 6.8 \cdot 10^{-44}:\\ \;\;\;\;1 \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 1.0 (exp (* M (- M))))))
   (if (<= M -1000000.0)
     t_0
     (if (<= M 6.8e-44)
       (* 1.0 (exp (* (* m m) -0.25)))
       (if (<= M 26.5) (* 1.0 (exp (- l))) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * exp((M * -M));
	double tmp;
	if (M <= -1000000.0) {
		tmp = t_0;
	} else if (M <= 6.8e-44) {
		tmp = 1.0 * exp(((m * m) * -0.25));
	} else if (M <= 26.5) {
		tmp = 1.0 * exp(-l);
	} else {
		tmp = t_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 = 1.0d0 * exp((m_1 * -m_1))
    if (m_1 <= (-1000000.0d0)) then
        tmp = t_0
    else if (m_1 <= 6.8d-44) then
        tmp = 1.0d0 * exp(((m * m) * (-0.25d0)))
    else if (m_1 <= 26.5d0) then
        tmp = 1.0d0 * exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * Math.exp((M * -M));
	double tmp;
	if (M <= -1000000.0) {
		tmp = t_0;
	} else if (M <= 6.8e-44) {
		tmp = 1.0 * Math.exp(((m * m) * -0.25));
	} else if (M <= 26.5) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = 1.0 * math.exp((M * -M))
	tmp = 0
	if M <= -1000000.0:
		tmp = t_0
	elif M <= 6.8e-44:
		tmp = 1.0 * math.exp(((m * m) * -0.25))
	elif M <= 26.5:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(1.0 * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (M <= -1000000.0)
		tmp = t_0;
	elseif (M <= 6.8e-44)
		tmp = Float64(1.0 * exp(Float64(Float64(m * m) * -0.25)));
	elseif (M <= 26.5)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = 1.0 * exp((M * -M));
	tmp = 0.0;
	if (M <= -1000000.0)
		tmp = t_0;
	elseif (M <= 6.8e-44)
		tmp = 1.0 * exp(((m * m) * -0.25));
	elseif (M <= 26.5)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(1.0 * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1000000.0], t$95$0, If[LessEqual[M, 6.8e-44], N[(1.0 * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 26.5], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1e6 or 26.5 < M

   1. Initial program 83.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 25.1

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 25.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.3%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

       \[\leadsto 1 \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

       5. lower-neg.f64 99.2

          \[\leadsto 1 \cdot e^{M \cdot \color{blue}{\left(-M\right)}} \]

   14. Applied rewrites 99.2%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if -1e6 < M < 6.80000000000000033e-44

   1. Initial program 67.7%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 38.1

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 38.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.1%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 52.3

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

   14. Applied rewrites 52.3%

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

   if 6.80000000000000033e-44 < M < 26.5

   1. Initial program 84.7%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 76.8

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 76.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 76.8%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 4.4 \cdot 10^{-203}:\\ \;\;\;\;1 \cdot e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 450:\\ \;\;\;\;1 \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 4.4e-203)
   (* 1.0 (exp (* (* m m) -0.25)))
   (if (<= n 450.0)
     (* 1.0 (exp (* M (- M))))
     (* 1.0 (exp (* (* n n) -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 4.4e-203) {
		tmp = 1.0 * exp(((m * m) * -0.25));
	} else if (n <= 450.0) {
		tmp = 1.0 * exp((M * -M));
	} else {
		tmp = 1.0 * exp(((n * n) * -0.25));
	}
	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) :: tmp
    if (n <= 4.4d-203) then
        tmp = 1.0d0 * exp(((m * m) * (-0.25d0)))
    else if (n <= 450.0d0) then
        tmp = 1.0d0 * exp((m_1 * -m_1))
    else
        tmp = 1.0d0 * exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 4.4e-203) {
		tmp = 1.0 * Math.exp(((m * m) * -0.25));
	} else if (n <= 450.0) {
		tmp = 1.0 * Math.exp((M * -M));
	} else {
		tmp = 1.0 * Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 4.4e-203:
		tmp = 1.0 * math.exp(((m * m) * -0.25))
	elif n <= 450.0:
		tmp = 1.0 * math.exp((M * -M))
	else:
		tmp = 1.0 * math.exp(((n * n) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 4.4e-203)
		tmp = Float64(1.0 * exp(Float64(Float64(m * m) * -0.25)));
	elseif (n <= 450.0)
		tmp = Float64(1.0 * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(1.0 * exp(Float64(Float64(n * n) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 4.4e-203)
		tmp = 1.0 * exp(((m * m) * -0.25));
	elseif (n <= 450.0)
		tmp = 1.0 * exp((M * -M));
	else
		tmp = 1.0 * exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 4.4e-203], N[(1.0 * N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 450.0], N[(1.0 * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 4.3999999999999999e-203

   1. Initial program 75.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 34.6

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 34.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.9%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in m around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {m}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 50.9

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

   14. Applied rewrites 50.9%

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

   if 4.3999999999999999e-203 < n < 450

   1. Initial program 91.7%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 35.5

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 35.5%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.5%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

       \[\leadsto 1 \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

       5. lower-neg.f64 68.0

          \[\leadsto 1 \cdot e^{M \cdot \color{blue}{\left(-M\right)}} \]

   14. Applied rewrites 68.0%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if 450 < n

   1. Initial program 61.8%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 31.1

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 31.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 31.1%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in n around inf

       \[\leadsto 1 \cdot e^{\color{blue}{\frac{-1}{4} \cdot {n}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot \frac{-1}{4}}} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot \frac{-1}{4}} \]

       4. lower-*.f64 98.2

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

   14. Applied rewrites 98.2%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -3.25 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 26.5:\\ \;\;\;\;1 \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 1.0 (exp (* M (- M))))))
   (if (<= M -3.25e-9) t_0 (if (<= M 26.5) (* 1.0 (exp (- l))) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * exp((M * -M));
	double tmp;
	if (M <= -3.25e-9) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = 1.0 * exp(-l);
	} else {
		tmp = t_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 = 1.0d0 * exp((m_1 * -m_1))
    if (m_1 <= (-3.25d-9)) then
        tmp = t_0
    else if (m_1 <= 26.5d0) then
        tmp = 1.0d0 * exp(-l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 * Math.exp((M * -M));
	double tmp;
	if (M <= -3.25e-9) {
		tmp = t_0;
	} else if (M <= 26.5) {
		tmp = 1.0 * Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = 1.0 * math.exp((M * -M))
	tmp = 0
	if M <= -3.25e-9:
		tmp = t_0
	elif M <= 26.5:
		tmp = 1.0 * math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(1.0 * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (M <= -3.25e-9)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = Float64(1.0 * exp(Float64(-l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = 1.0 * exp((M * -M));
	tmp = 0.0;
	if (M <= -3.25e-9)
		tmp = t_0;
	elseif (M <= 26.5)
		tmp = 1.0 * exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(1.0 * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -3.25e-9], t$95$0, If[LessEqual[M, 26.5], N[(1.0 * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -3.2500000000000002e-9 or 26.5 < M

   1. Initial program 83.3%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 24.6

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 24.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.7%

       \[\leadsto 1 \cdot e^{-\ell} \]

   12. Taylor expanded in M around inf

       \[\leadsto 1 \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot e^{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}} \]

       2. unpow2 N/A

          \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\color{blue}{M \cdot M}\right)} \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(\mathsf{neg}\left(M\right)\right)}} \]

       5. lower-neg.f64 97.6

          \[\leadsto 1 \cdot e^{M \cdot \color{blue}{\left(-M\right)}} \]

   14. Applied rewrites 97.6%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if -3.2500000000000002e-9 < M < 26.5

   1. Initial program 69.2%

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

   3. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in l around inf

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

      2. lower-neg.f64 42.3

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   8. Applied rewrites 42.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   9. Taylor expanded in M around 0

      \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 42.3%

       \[\leadsto 1 \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 34.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot e^{-\ell} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 * exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 * exp(-l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. lower-cos.f64 94.7

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

5. Applied rewrites 94.7%

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

6. Taylor expanded in l around inf

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{neg}\left(\ell\right)}} \]

   2. lower-neg.f64 34.0

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

8. Applied rewrites 34.0%

   \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

9. Taylor expanded in M around 0

   \[\leadsto 1 \cdot e^{\mathsf{neg}\left(\ell\right)} \]
10. Step-by-step derivation

11. Applied rewrites 33.6%

    \[\leadsto 1 \cdot e^{-\ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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)))))))