Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.8%

Time: 23.8s

Alternatives: 7

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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: 96.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 96.7%

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

      1. add-log-exp 96.7%

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

      2. associate-/l* 96.7%

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

      3. fma-neg 96.7%

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

      4. div-inv 96.7%

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

      5. metadata-eval 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in l around inf 96.7%

      \[\leadsto \log \left(e^{\cos \left(\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)\right)}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\ell}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 96.7%

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

      2. pow3 96.7%

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

      3. add-sqr-sqrt 55.9%

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

      4. sqrt-unprod 69.8%

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

      5. sqr-neg 69.8%

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

      6. sqrt-unprod 40.8%

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

      7. add-sqr-sqrt 96.7%

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

   7. Applied egg-rr 96.7%

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

   if 0.5 < (*.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 29.7%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 97.3%

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

Alternative 2: 96.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 96.7%

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

      1. add-log-exp 96.7%

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

      2. associate-/l* 96.7%

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

      3. fma-neg 96.7%

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

      4. div-inv 96.7%

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

      5. metadata-eval 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in l around inf 96.7%

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

   if 0.5 < (*.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 29.7%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 97.3%

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

Alternative 3: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((((k * (m + n)) / 2.0d0) - m_1))
    t_1 = (((m + n) / 2.0d0) - m_1) ** 2.0d0
    t_2 = exp(((abs((m - n)) - l) - t_1))
    if ((t_0 * t_2) <= 0.5d0) then
        tmp = t_0 * exp((-l - t_1))
    else
        tmp = t_2 * cos(m_1)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 96.7%

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

   3. Taylor expanded in l around inf 96.7%

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

   if 0.5 < (*.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 29.7%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 97.3%

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

Alternative 4: 96.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

5. Simplified 95.5%

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

6. Taylor expanded in l around inf 95.4%

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

7. Final simplification 95.4%

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

Alternative 5: 86.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+112}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2e+112)
   (exp (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l))
   (* (cos M) (exp (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2e+112) {
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l));
	}
	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 <= (-2d+112)) then
        tmp = exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) - l))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - l))
    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 <= -2e+112) {
		tmp = Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2e+112:
		tmp = math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2e+112)
		tmp = exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) - l));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2e+112)
		tmp = exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2e+112], N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.9999999999999999e112

   1. Initial program 70.0%

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

   3. Taylor expanded in l around inf 70.0%

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

   4. Taylor expanded in n around 0 47.6%

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

      1. +-commutative 47.6%

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

      2. unpow2 47.6%

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

      3. distribute-rgt-out 57.7%

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

      4. *-commutative 57.7%

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

      5. *-commutative 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in m around inf 57.7%

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

      1. associate-*r* 57.7%

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

   9. Simplified 57.7%

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

   10. Taylor expanded in K around 0 87.7%

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

   if -1.9999999999999999e112 < m

   1. Initial program 78.7%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in l around inf 94.5%

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

   7. Taylor expanded in m around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. unpow2 80.3%

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

      3. distribute-rgt-out 85.4%

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

      4. *-commutative 85.4%

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

      5. *-commutative 85.4%

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

   9. Simplified 85.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+112}:\\ \;\;\;\;e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.45 \cdot 10^{+105}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.45e+105)
   (* (cos M) (exp (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l)))
   (* (cos M) (exp (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.45e+105) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1.45d+105) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) - l))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.45e+105) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.45e+105:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.45e+105)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.45e+105)
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.45e+105], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.45000000000000005e105

   1. Initial program 79.4%

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

   3. Taylor expanded in l around inf 79.2%

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

   4. Taylor expanded in n around 0 66.4%

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

      1. +-commutative 66.4%

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

      2. unpow2 66.4%

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

      3. distribute-rgt-out 73.1%

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

      4. *-commutative 73.1%

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

      5. *-commutative 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in K around 0 84.9%

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

   9. Simplified 84.9%

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

   if 1.45000000000000005e105 < n

   1. Initial program 68.1%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 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. Simplified 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 100.0%

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

   7. Taylor expanded in m around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. distribute-rgt-out 95.8%

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

      4. *-commutative 95.8%

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

      5. *-commutative 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.45 \cdot 10^{+105}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in l around inf 77.2%

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

4. Taylor expanded in n around 0 59.7%

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

   1. +-commutative 59.7%

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

   2. unpow2 59.7%

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

   3. distribute-rgt-out 67.2%

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

   4. *-commutative 67.2%

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

   5. *-commutative 67.2%

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

6. Simplified 67.2%

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

7. Taylor expanded in m around inf 68.9%

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

   1. associate-*r* 68.9%

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

9. Simplified 68.9%

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

10. Taylor expanded in K around 0 80.2%

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

11. Final simplification 80.2%

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

Reproduce

herbie shell --seed 2024072 
(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)))))))