Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.5% → 96.9%

Time: 15.9s

Alternatives: 7

Speedup: 3.5×

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: 75.5% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5\\ t_1 := \sqrt[3]{t\_0}\\ t_2 := e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}\\ t_3 := t\_0 - M\\ \mathbf{if}\;\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot t\_2 \leq \infty:\\ \;\;\;\;t\_2 \cdot \cos \left(t\_1 \cdot \left(K \cdot {t\_1}^{2}\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m - n\right) - \left(\ell + t\_3 \cdot t\_3\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * 0.5)
	t_1 = cbrt(t_0)
	t_2 = exp(Float64(Float64(abs(Float64(m - n)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)))
	t_3 = Float64(t_0 - M)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * t_2) <= Inf)
		tmp = Float64(t_2 * cos(Float64(Float64(t_1 * Float64(K * (t_1 ^ 2.0))) - M)));
	else
		tmp = exp(Float64(Float64(m - n) - Float64(l + Float64(t_3 * t_3))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - M), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], Infinity], N[(t$95$2 * N[Cos[N[(N[(t$95$1 * N[(K * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(m - n), $MachinePrecision] - N[(l + N[(t$95$3 * t$95$3), $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 94.0%

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

      1. associate-/l* 94.0%

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

      2. add-cube-cbrt 96.5%

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

      3. associate-*r* 96.5%

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

      4. pow2 96.5%

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

      5. div-inv 96.5%

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

      6. metadata-eval 96.5%

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

      7. div-inv 96.5%

         \[\leadsto \cos \left(\left(K \cdot {\left(\sqrt[3]{\left(m + n\right) \cdot 0.5}\right)}^{2}\right) \cdot \sqrt[3]{\color{blue}{\left(m + n\right) \cdot \frac{1}{2}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      8. metadata-eval 96.5%

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

   4. Applied egg-rr 96.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in K around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 100.0%

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

      1. unpow2 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in m around -inf 100.0%

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

       1. fabs-neg 100.0%

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

       2. mul-1-neg 100.0%

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

       3. sub-neg 100.0%

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

       4. fabs-sub 100.0%

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

       5. rem-square-sqrt 50.0%

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

       6. fabs-sqr 50.0%

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

       7. rem-square-sqrt 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 97.1%

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

Alternative 2: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((abs((m - n)) - (l + ((((m + n) * 0.5d0) - m_1) ** 2.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp((Math.abs((m - n)) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in K around 0 95.3%

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

5. Simplified 95.3%

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

Alternative 3: 96.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return exp((fabs((m - n)) - (l + (t_0 * t_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
    real(8) :: t_0
    t_0 = ((m + n) * 0.5d0) - m_1
    code = exp((abs((m - n)) - (l + (t_0 * t_0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in K around 0 95.3%

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

5. Simplified 95.3%

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

6. Taylor expanded in M around 0 95.1%

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

   1. unpow2 95.1%

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

8. Applied egg-rr 95.1%

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

9. Final simplification 95.1%

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

Alternative 4: 96.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) * 0.5) - M;
	return exp(((m - n) - (l + (t_0 * t_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
    real(8) :: t_0
    t_0 = ((m + n) * 0.5d0) - m_1
    code = exp(((m - n) - (l + (t_0 * t_0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in K around 0 95.3%

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

5. Simplified 95.3%

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

6. Taylor expanded in M around 0 95.1%

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

   1. unpow2 95.1%

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

8. Applied egg-rr 95.1%

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

9. Taylor expanded in m around -inf 95.1%

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

    1. fabs-neg 95.1%

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

    2. mul-1-neg 95.1%

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

    3. sub-neg 95.1%

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

    4. fabs-sub 95.1%

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

    5. rem-square-sqrt 45.8%

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

    6. fabs-sqr 45.8%

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

    7. rem-square-sqrt 94.5%

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

11. Simplified 94.5%

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

12. Final simplification 94.5%

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

Alternative 5: 35.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in K around 0 35.4%

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

   1. cos-neg 35.4%

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

   2. associate-*r* 35.4%

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

   3. sin-neg 35.4%

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

8. Simplified 35.4%

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

9. Taylor expanded in M around 0 37.2%

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

Alternative 6: 6.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in l around 0 7.1%

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

7. Taylor expanded in K around 0 7.6%

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

   1. cos-neg 7.6%

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

9. Simplified 7.6%

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

Alternative 7: 6.9% accurate, 425.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in l around inf 33.0%

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

   1. mul-1-neg 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in l around 0 7.1%

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

7. Taylor expanded in K around 0 7.6%

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

   1. cos-neg 7.6%

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

9. Simplified 7.6%

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

10. Taylor expanded in M around 0 7.6%

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

Reproduce

herbie shell --seed 2024132 
(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)))))))