Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.9%

Time: 13.7s

Alternatives: 9

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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: 96.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ m n)) M)))
   (* (cos M) (exp (* l (- -1.0 (/ (- (* t_0 t_0) (fabs (- m n))) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (m + n)) - M;
	return cos(M) * exp((l * (-1.0 - (((t_0 * t_0) - fabs((m - n))) / l))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = (0.5d0 * (m + n)) - m_1
    code = cos(m_1) * exp((l * ((-1.0d0) - (((t_0 * t_0) - abs((m - n))) / l))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (m + n)) - M;
	return Math.cos(M) * Math.exp((l * (-1.0 - (((t_0 * t_0) - Math.abs((m - n))) / l))));
}

Python

def code(K, m, n, M, l):
	t_0 = (0.5 * (m + n)) - M
	return math.cos(M) * math.exp((l * (-1.0 - (((t_0 * t_0) - math.fabs((m - n))) / l))))

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(m + n)) - M)
	return Float64(cos(M) * exp(Float64(l * Float64(-1.0 - Float64(Float64(Float64(t_0 * t_0) - abs(Float64(m - n))) / l)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	t_0 = (0.5 * (m + n)) - M;
	tmp = cos(M) * exp((l * (-1.0 - (((t_0 * t_0) - abs((m - n))) / l))));
end

Wolfram

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

Derivation

1. Initial program 75.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 95.9

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in l around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-neg.f64 N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 95.9%

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

9. Final simplification 95.9%

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

Alternative 2: 96.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(m + n\right) - M\\ e^{\ell \cdot \left(-1 - \frac{t\_0 \cdot t\_0 - \left|m - n\right|}{\ell}\right)} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (+ m n)) M)))
   (exp (* l (- -1.0 (/ (- (* t_0 t_0) (fabs (- m n))) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (m + n)) - M;
	return exp((l * (-1.0 - (((t_0 * t_0) - fabs((m - n))) / l))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = (0.5d0 * (m + n)) - m_1
    code = exp((l * ((-1.0d0) - (((t_0 * t_0) - abs((m - n))) / l))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = (0.5 * (m + n)) - M;
	return Math.exp((l * (-1.0 - (((t_0 * t_0) - Math.abs((m - n))) / l))));
}

Python

def code(K, m, n, M, l):
	t_0 = (0.5 * (m + n)) - M
	return math.exp((l * (-1.0 - (((t_0 * t_0) - math.fabs((m - n))) / l))))

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(0.5 * Float64(m + n)) - M)
	return exp(Float64(l * Float64(-1.0 - Float64(Float64(Float64(t_0 * t_0) - abs(Float64(m - n))) / l))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	t_0 = (0.5 * (m + n)) - M;
	tmp = exp((l * (-1.0 - (((t_0 * t_0) - abs((m - n))) / l))));
end

Wolfram

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

Derivation

1. Initial program 75.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 95.9

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in l around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-neg.f64 N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 95.9%

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

9. Taylor expanded in M around 0

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

11. Applied rewrites 95.8%

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

12. Final simplification 95.8%

    \[\leadsto e^{\ell \cdot \left(-1 - \frac{\left(0.5 \cdot \left(m + n\right) - M\right) \cdot \left(0.5 \cdot \left(m + n\right) - M\right) - \left|m - n\right|}{\ell}\right)} \]
13. Add Preprocessing

Alternative 3: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -2.8 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 5 \cdot 10^{+41}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -2.8e+101)
     t_0
     (if (<= M 5e+41)
       (exp (- (fabs (- m n)) (fma 0.25 (* (+ m n) (+ m n)) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -2.8e+101) {
		tmp = t_0;
	} else if (M <= 5e+41) {
		tmp = exp((fabs((m - n)) - fma(0.25, ((m + n) * (m + n)), l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -2.8e+101)
		tmp = t_0;
	elseif (M <= 5e+41)
		tmp = exp(Float64(abs(Float64(m - n)) - fma(0.25, Float64(Float64(m + n) * Float64(m + n)), l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -2.8e+101], t$95$0, If[LessEqual[M, 5e+41], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -2.79999999999999981e101 or 5.00000000000000022e41 < M

   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

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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

   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 100.0

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

   14. Applied rewrites 100.0%

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

   if -2.79999999999999981e101 < M < 5.00000000000000022e41

   1. Initial program 72.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in M around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto e^{\left|m - n\right| - \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(m + n\right)}^{2}, \ell\right)}} \]

      7. unpow2 N/A

         \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(m + n\right) \cdot \left(m + n\right)}, \ell\right)} \]

      9. +-commutative N/A

         \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(n + m\right)} \cdot \left(m + n\right), \ell\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{1}{4}, \left(n + m\right) \cdot \color{blue}{\left(n + m\right)}, \ell\right)} \]

      12. lower-+.f64 92.1

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

   8. Applied rewrites 92.1%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(n + m\right) \cdot \left(n + m\right), \ell\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.8 \cdot 10^{+101}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 5 \cdot 10^{+41}:\\ \;\;\;\;e^{\left|m - n\right| - \mathsf{fma}\left(0.25, \left(m + n\right) \cdot \left(m + n\right), \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.34:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-255}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.34)
   (exp (* (* m m) -0.25))
   (if (<= m -1e-255) (exp (* M (- M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.34) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -1e-255) {
		tmp = exp((M * -M));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	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 (m <= (-0.34d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-1d-255)) then
        tmp = exp((m_1 * -m_1))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.34) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -1e-255) {
		tmp = Math.exp((M * -M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.34:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -1e-255:
		tmp = math.exp((M * -M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.34)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -1e-255)
		tmp = exp(Float64(M * Float64(-M)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.34)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -1e-255)
		tmp = exp((M * -M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.34], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1e-255], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.340000000000000024

   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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 98.7%

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

   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 97.5

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

   14. Applied rewrites 97.5%

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

   if -0.340000000000000024 < m < -1e-255

   1. Initial program 84.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 95.8%

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

   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 63.5

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

   14. Applied rewrites 63.5%

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

   if -1e-255 < m

   1. Initial program 76.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 94.2%

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

   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 55.0

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

   14. Applied rewrites 55.0%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.34:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -1 \cdot 10^{-255}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -1.6 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -1.6e+21) t_0 (if (<= M 27.0) (exp (* (* m m) -0.25)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -1.6e+21) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = exp(((m * m) * -0.25));
	} 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 = exp((m_1 * -m_1))
    if (m_1 <= (-1.6d+21)) then
        tmp = t_0
    else if (m_1 <= 27.0d0) then
        tmp = exp(((m * m) * (-0.25d0)))
    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 = Math.exp((M * -M));
	double tmp;
	if (M <= -1.6e+21) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((M * -M))
	tmp = 0
	if M <= -1.6e+21:
		tmp = t_0
	elif M <= 27.0:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -1.6e+21)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((M * -M));
	tmp = 0.0;
	if (M <= -1.6e+21)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -1.6e+21], t$95$0, If[LessEqual[M, 27.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -1.6e21 or 27 < M

   1. Initial program 83.2%

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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

   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 98.5

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

   14. Applied rewrites 98.5%

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

   if -1.6e21 < M < 27

   1. Initial program 67.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in l around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.7%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 91.4%

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

   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 53.5

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

   14. Applied rewrites 53.5%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.6 \cdot 10^{+21}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -2.1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 4.5 \cdot 10^{-63}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -2.1e+17) t_0 (if (<= M 4.5e-63) (exp (- l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -2.1e+17) {
		tmp = t_0;
	} else if (M <= 4.5e-63) {
		tmp = 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 = exp((m_1 * -m_1))
    if (m_1 <= (-2.1d+17)) then
        tmp = t_0
    else if (m_1 <= 4.5d-63) then
        tmp = 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 = Math.exp((M * -M));
	double tmp;
	if (M <= -2.1e+17) {
		tmp = t_0;
	} else if (M <= 4.5e-63) {
		tmp = Math.exp(-l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((M * -M))
	tmp = 0
	if M <= -2.1e+17:
		tmp = t_0
	elif M <= 4.5e-63:
		tmp = math.exp(-l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -2.1e+17)
		tmp = t_0;
	elseif (M <= 4.5e-63)
		tmp = exp(Float64(-l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((M * -M));
	tmp = 0.0;
	if (M <= -2.1e+17)
		tmp = t_0;
	elseif (M <= 4.5e-63)
		tmp = exp(-l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -2.1e+17], t$95$0, If[LessEqual[M, 4.5e-63], N[Exp[(-l)], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -2.1e17 or 4.5e-63 < M

   1. Initial program 83.0%

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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in M around 0

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

   11. Applied rewrites 100.0%

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

   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 93.8

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

   14. Applied rewrites 93.8%

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

   if -2.1e17 < M < 4.5e-63

   1. Initial program 66.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in l around inf

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 34.8

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

   8. Applied rewrites 34.8%

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

   9. Taylor expanded in M around 0

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

       1. neg-mul-1 N/A

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

       2. lower-exp.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 34.8

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

   11. Applied rewrites 34.8%

       \[\leadsto \color{blue}{e^{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.1 \cdot 10^{+17}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 4.5 \cdot 10^{-63}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 95.9

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in l around inf

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

   1. neg-mul-1 N/A

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

   2. lower-neg.f64 29.6

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

8. Applied rewrites 29.6%

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

9. Taylor expanded in M around 0

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

    1. neg-mul-1 N/A

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

    2. lower-exp.f64 N/A

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

    3. neg-mul-1 N/A

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

    4. lower-neg.f64 29.6

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

11. Applied rewrites 29.6%

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

Alternative 8: 7.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

3. Taylor expanded in m around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in m around 0

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

   1. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot n\right) - M\right)} \]

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, K \cdot n, \mathsf{neg}\left(M\right)\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{n \cdot K}, \mathsf{neg}\left(M\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{n \cdot K}, \mathsf{neg}\left(M\right)\right)\right) \]

   6. lower-neg.f64 5.0

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

8. Applied rewrites 5.0%

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

9. Taylor expanded in n around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 5.4

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

11. Applied rewrites 5.4%

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

Alternative 9: 6.9% accurate, 359.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 75.7%

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

3. Taylor expanded in m around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in m around 0

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

   1. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{2} \cdot \left(K \cdot n\right) - M\right)} \]

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, K \cdot n, \mathsf{neg}\left(M\right)\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{n \cdot K}, \mathsf{neg}\left(M\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{n \cdot K}, \mathsf{neg}\left(M\right)\right)\right) \]

   6. lower-neg.f64 5.0

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

8. Applied rewrites 5.0%

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

9. Taylor expanded in n around 0

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

    1. cos-neg N/A

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

    2. lower-cos.f64 5.4

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

11. Applied rewrites 5.4%

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

12. Taylor expanded in M around 0

    \[\leadsto \color{blue}{1} \]
13. Step-by-step derivation

14. Applied rewrites 5.4%

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

Reproduce

herbie shell --seed 2024216 
(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)))))))