2atan (example 3.5)

Percentage Accurate: 8.5% → 99.9%

Time: 9.3s

Alternatives: 7

Speedup: 1.9×

Specification

\[N > 1 \land N < 10^{+100}\]

Math

\[\begin{array}{l} \\ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan((n + 1.0d0)) - atan(n)
end function

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

Julia

function code(N)
	return Float64(atan(Float64(N + 1.0)) - atan(N))
end

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

Wolfram

code[N_] := N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 8.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan((n + 1.0d0)) - atan(n)
end function

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

Julia

function code(N)
	return Float64(atan(Float64(N + 1.0)) - atan(N))
end

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

Wolfram

code[N_] := N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left({\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}^{-0.5}\right)}^{-2} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (pow (pow (atan2 1.0 (+ 1.0 (fma N N N))) -0.5) -2.0))

C

double code(double N) {
	return pow(pow(atan2(1.0, (1.0 + fma(N, N, N))), -0.5), -2.0);
}

Julia

function code(N)
	return (atan(1.0, Float64(1.0 + fma(N, N, N))) ^ -0.5) ^ -2.0
end

Wolfram

code[N_] := N[Power[N[Power[N[ArcTan[1.0 / N[(1.0 + N[(N * N + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -0.5], $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 10.1%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}}}} \]

   6. flip-- N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   8. lower-/.f64 10.1

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   10. lift-atan.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N}} \]

   11. lift-atan.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N}}} \]

   12. diff-atan N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}}}} \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{\left(N + 1\right)} - N}{1 + \left(N + 1\right) \cdot N}}} \]

   14. +-commutative N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{\left(1 + N\right)} - N}{1 + \left(N + 1\right) \cdot N}}} \]

   15. associate--l+ N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{1 + \left(N - N\right)}}{1 + \left(N + 1\right) \cdot N}}} \]

   16. +-inverses N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{1 + \color{blue}{0}}{1 + \left(N + 1\right) \cdot N}}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{1}}{1 + \left(N + 1\right) \cdot N}}} \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{1}{\color{blue}{\left(N + 1\right) \cdot N + 1}}}} \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}}} \]

   2. inv-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{-1}}} \]

   3. sqr-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)} \cdot {\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}}} \]

   4. pow2 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{{\color{blue}{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}\right)}}^{2}} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, \color{blue}{N + 1}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   8. +-commutative N/A

      \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   10. metadata-eval 99.6

       \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\color{blue}{-0.5}}\right)}^{2}} \]

6. Applied rewrites 99.6%

   \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{-0.5}\right)}^{2}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{2}}} \]

   3. pow-flip N/A

      \[\leadsto \color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}} \]

   5. lift-fma.f64 N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{N \cdot N + \left(1 + N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{\left(1 + N\right) + N \cdot N}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{\left(1 + N\right)} + N \cdot N}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   8. associate-+r+ N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{1 + \left(N + N \cdot N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{1 + \color{blue}{\left(N \cdot N + N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   10. lift-fma.f64 N/A

       \[\leadsto {\left({\tan^{-1}_* \frac{1}{1 + \color{blue}{\mathsf{fma}\left(N, N, N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{1 + \mathsf{fma}\left(N, N, N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   12. metadata-eval 99.9

       \[\leadsto {\left({\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}^{-0.5}\right)}^{\color{blue}{-2}} \]

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{{\left({\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}^{-0.5}\right)}^{-2}} \]
9. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}}\right)}^{-2} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (pow (sqrt (/ 1.0 (atan2 1.0 (+ 1.0 (fma N N N))))) -2.0))

C

double code(double N) {
	return pow(sqrt((1.0 / atan2(1.0, (1.0 + fma(N, N, N))))), -2.0);
}

Julia

function code(N)
	return sqrt(Float64(1.0 / atan(1.0, Float64(1.0 + fma(N, N, N))))) ^ -2.0
end

Wolfram

code[N_] := N[Power[N[Sqrt[N[(1.0 / N[ArcTan[1.0 / N[(1.0 + N[(N * N + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\tan^{-1} \left(N + 1\right) \cdot \tan^{-1} \left(N + 1\right) - \tan^{-1} N \cdot \tan^{-1} N}{\tan^{-1} \left(N + 1\right) + \tan^{-1} N}}}} \]

   6. flip-- N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   8. lower-/.f64 8.5

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N}}} \]

   10. lift-atan.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N}} \]

   11. lift-atan.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N}}} \]

   12. diff-atan N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}}}} \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{\left(N + 1\right)} - N}{1 + \left(N + 1\right) \cdot N}}} \]

   14. +-commutative N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{\left(1 + N\right)} - N}{1 + \left(N + 1\right) \cdot N}}} \]

   15. associate--l+ N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{1 + \left(N - N\right)}}{1 + \left(N + 1\right) \cdot N}}} \]

   16. +-inverses N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{1 + \color{blue}{0}}{1 + \left(N + 1\right) \cdot N}}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{\color{blue}{1}}{1 + \left(N + 1\right) \cdot N}}} \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{1}{\color{blue}{\left(N + 1\right) \cdot N + 1}}}} \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}}} \]

   2. inv-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{-1}}} \]

   3. sqr-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)} \cdot {\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}}} \]

   4. pow2 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{{\color{blue}{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}^{\left(\frac{-1}{2}\right)}\right)}}^{2}} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, \color{blue}{N + 1}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   8. +-commutative N/A

      \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   10. metadata-eval 99.5

       \[\leadsto \frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\color{blue}{-0.5}}\right)}^{2}} \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{-0.5}\right)}^{2}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{2}}} \]

   3. pow-flip N/A

      \[\leadsto \color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\left({\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}} \]

   5. lift-fma.f64 N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{N \cdot N + \left(1 + N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{\left(1 + N\right) + N \cdot N}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{\left(1 + N\right)} + N \cdot N}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   8. associate-+r+ N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{1 + \left(N + N \cdot N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto {\left({\tan^{-1}_* \frac{1}{1 + \color{blue}{\left(N \cdot N + N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   10. lift-fma.f64 N/A

       \[\leadsto {\left({\tan^{-1}_* \frac{1}{1 + \color{blue}{\mathsf{fma}\left(N, N, N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto {\left({\tan^{-1}_* \frac{1}{\color{blue}{1 + \mathsf{fma}\left(N, N, N\right)}}}^{\frac{-1}{2}}\right)}^{\left(\mathsf{neg}\left(2\right)\right)} \]

   12. metadata-eval 99.9

       \[\leadsto {\left({\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}^{-0.5}\right)}^{\color{blue}{-2}} \]

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{{\left({\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}^{-0.5}\right)}^{-2}} \]

9. Taylor expanded in N around 0

   \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \left(N + {N}^{2}\right)}}}\right)}}^{-2} \]
10. Step-by-step derivation

    1. lower-sqrt.f64 N/A

       \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \left(N + {N}^{2}\right)}}}\right)}}^{-2} \]

    2. lower-/.f64 N/A

       \[\leadsto {\left(\sqrt{\color{blue}{\frac{1}{\tan^{-1}_* \frac{1}{1 + \left(N + {N}^{2}\right)}}}}\right)}^{-2} \]

    3. lower-atan2.f64 N/A

       \[\leadsto {\left(\sqrt{\frac{1}{\color{blue}{\tan^{-1}_* \frac{1}{1 + \left(N + {N}^{2}\right)}}}}\right)}^{-2} \]

    4. lower-+.f64 N/A

       \[\leadsto {\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{\color{blue}{1 + \left(N + {N}^{2}\right)}}}}\right)}^{-2} \]

    5. +-commutative N/A

       \[\leadsto {\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \color{blue}{\left({N}^{2} + N\right)}}}}\right)}^{-2} \]

    6. unpow2 N/A

       \[\leadsto {\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \left(\color{blue}{N \cdot N} + N\right)}}}\right)}^{-2} \]

    7. lower-fma.f64 99.8

       \[\leadsto {\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \color{blue}{\mathsf{fma}\left(N, N, N\right)}}}}\right)}^{-2} \]

11. Applied rewrites 99.8%

    \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}}}\right)}}^{-2} \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0))))))

C

double code(double N) {
	return atan((1.0 / (1.0 + (N * (N + 1.0)))));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan((1.0d0 / (1.0d0 + (n * (n + 1.0d0)))))
end function

Java

public static double code(double N) {
	return Math.atan((1.0 / (1.0 + (N * (N + 1.0)))));
}

Python

def code(N):
	return math.atan((1.0 / (1.0 + (N * (N + 1.0)))))

Julia

function code(N)
	return atan(Float64(1.0 / Float64(1.0 + Float64(N * Float64(N + 1.0)))))
end

MATLAB

function tmp = code(N)
	tmp = atan((1.0 / (1.0 + (N * (N + 1.0)))));
end

Wolfram

code[N_] := N[ArcTan[N[(1.0 / N[(1.0 + N[(N * N[(N + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Developer Target 2: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, 1 + N, 1\right)} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (fma N (+ 1.0 N) 1.0)))

C

double code(double N) {
	return atan2(1.0, fma(N, (1.0 + N), 1.0));
}

Julia

function code(N)
	return atan(1.0, fma(N, Float64(1.0 + N), 1.0))
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N * N[(1.0 + N), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024226 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+100))

  :alt
  (! :herbie-platform default (atan (/ 1 (+ 1 (* N (+ N 1))))))

  :alt
  (! :herbie-platform default (atan2 1 (fma N (+ 1 N) 1)))

  (- (atan (+ N 1.0)) (atan N)))