Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.8%

Time: 11.8s

Alternatives: 8

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

3. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fabs.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 97.7%

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

6. Final simplification 97.7%

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

Alternative 2: 96.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fabs.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 96.8%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 96.4%

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

9. Final simplification 96.4%

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

Reproduce

herbie shell --seed 2024227 
(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)))))))