2atan (example 3.5)

Percentage Accurate: 9.0% → 99.6%

Time: 8.0s

Alternatives: 4

Speedup: 2.0×

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: 9.0% 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: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan2(1.0d0, (1.0d0 + (n * (1.0d0 + n))))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, (1.0 + (N * (1.0 + N))));
}

Python

def code(N):
	return math.atan2(1.0, (1.0 + (N * (1.0 + N))))

Julia

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

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, (1.0 + (N * (1.0 + N))));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. diff-atan N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. associate-+r+ N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. atan2-lowering-atan2.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{\left(N \cdot N + \left(1 \cdot 1 + N \cdot 1\right)\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + \color{blue}{N} \cdot 1\right)\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + N\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(N + \color{blue}{1}\right)\right)\right) \]

   16. associate-+r+ N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N \cdot N + N\right) + \color{blue}{1}\right)\right) \]

   17. distribute-lft1-in N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N + 1\right) \cdot N + 1\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(1 + \color{blue}{\left(N + 1\right) \cdot N}\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(N + 1\right) \cdot N\right)}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \left(N \cdot \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   22. +-lowering-+.f64 99.7%

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \color{blue}{1}\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \tan^{-1}_* \frac{1}{1 + N \cdot \left(1 + N\right)} \]
6. Add Preprocessing

Alternative 2: 96.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan2(1.0d0, (n * (1.0d0 + n)))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, (N * (1.0 + N)));
}

Python

def code(N):
	return math.atan2(1.0, (N * (1.0 + N)))

Julia

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

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, (N * (1.0 + N)));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. diff-atan N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. associate-+r+ N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. atan2-lowering-atan2.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{\left(N \cdot N + \left(1 \cdot 1 + N \cdot 1\right)\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + \color{blue}{N} \cdot 1\right)\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + N\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(N + \color{blue}{1}\right)\right)\right) \]

   16. associate-+r+ N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N \cdot N + N\right) + \color{blue}{1}\right)\right) \]

   17. distribute-lft1-in N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N + 1\right) \cdot N + 1\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(1 + \color{blue}{\left(N + 1\right) \cdot N}\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(N + 1\right) \cdot N\right)}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \left(N \cdot \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   22. +-lowering-+.f64 99.7%

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \color{blue}{1}\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in N around inf

   \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{\left({N}^{2} \cdot \left(1 + \frac{1}{N}\right)\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N \cdot N\right) \cdot \left(\color{blue}{1} + \frac{1}{N}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot \color{blue}{\left(N \cdot \left(1 + \frac{1}{N}\right)\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot \left(N \cdot \left(\frac{1}{N} + \color{blue}{1}\right)\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot \left(N \cdot \frac{1}{N} + \color{blue}{N \cdot 1}\right)\right)\right) \]

   5. rgt-mult-inverse N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot \left(1 + \color{blue}{N} \cdot 1\right)\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot \left(1 + N\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(1 + N\right)}\right)\right) \]

   8. +-lowering-+.f64 96.9%

      \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{N}\right)\right)\right) \]

7. Simplified 96.9%

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

Alternative 3: 92.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{N \cdot N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (* N N)))

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan2(1.0d0, (n * n))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, (N * N));
}

Python

def code(N):
	return math.atan2(1.0, (N * N))

Julia

function code(N)
	return atan(1.0, Float64(N * N))
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, (N * N));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. diff-atan N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. associate-+r+ N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. atan2-lowering-atan2.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{\left(N \cdot N + \left(1 \cdot 1 + N \cdot 1\right)\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + \color{blue}{N} \cdot 1\right)\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + N\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(N + \color{blue}{1}\right)\right)\right) \]

   16. associate-+r+ N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N \cdot N + N\right) + \color{blue}{1}\right)\right) \]

   17. distribute-lft1-in N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N + 1\right) \cdot N + 1\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(1 + \color{blue}{\left(N + 1\right) \cdot N}\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(N + 1\right) \cdot N\right)}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \left(N \cdot \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   22. +-lowering-+.f64 99.7%

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \color{blue}{1}\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in N around inf

   \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{\left({N}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot \color{blue}{N}\right)\right) \]

   2. *-lowering-*.f64 93.9%

      \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{N}\right)\right) \]

7. Simplified 93.9%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot N}} \]
8. Add Preprocessing

Alternative 4: 6.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{1} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan2(1.0d0, 1.0d0)
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, 1.0);
}

Python

def code(N):
	return math.atan2(1.0, 1.0)

Julia

function code(N)
	return atan(1.0, 1.0)
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, 1.0);
end

Wolfram

code[N_] := N[ArcTan[1.0 / 1.0], $MachinePrecision]

Derivation

1. Initial program 8.2%

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

   1. diff-atan N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. associate-+r+ N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. atan2-lowering-atan2.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{\left(N \cdot N + \left(1 \cdot 1 + N \cdot 1\right)\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + \color{blue}{N} \cdot 1\right)\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(1 + N\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(N \cdot N + \left(N + \color{blue}{1}\right)\right)\right) \]

   16. associate-+r+ N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N \cdot N + N\right) + \color{blue}{1}\right)\right) \]

   17. distribute-lft1-in N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(\left(N + 1\right) \cdot N + 1\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \left(1 + \color{blue}{\left(N + 1\right) \cdot N}\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(N + 1\right) \cdot N\right)}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \left(N \cdot \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(N + 1\right)}\right)\right)\right) \]

   22. +-lowering-+.f64 99.7%

       \[\leadsto \mathsf{atan2.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \color{blue}{1}\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in N around 0

   \[\leadsto \mathsf{atan2.f64}\left(1, \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 6.4%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1}} \]
8. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.9× 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]

Reproduce

herbie shell --seed 2024161 
(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))))))

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