2atan (example 3.5)

Percentage Accurate: 9.0% → 99.6%

Time: 6.0s

Alternatives: 6

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: 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

   1. diff-atan 18.8%

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

   2. associate--l+ 18.8%

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

   3. +-commutative 18.8%

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

   4. *-commutative 18.8%

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

   5. fma-def 18.8%

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

4. Applied egg-rr 18.8%

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

   1. +-commutative 18.8%

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

   2. associate-+l- 99.5%

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

   3. +-inverses 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

6. Simplified 99.5%

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

   1. add-cube-cbrt 97.9%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)}}} \]

   2. pow3 97.9%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)}\right)}^{3}}} \]

8. Applied egg-rr 97.9%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)}\right)}^{3}}} \]
9. Step-by-step derivation

   1. unpow3 97.9%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(N, 1 + N, 1\right)}}} \]

   2. add-cube-cbrt 99.5%

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

   3. fma-udef 99.5%

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

   4. distribute-rgt-in 99.5%

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

   5. *-un-lft-identity 99.5%

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

   6. unpow2 99.5%

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

   7. associate-+l+ 99.5%

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

   8. unpow2 99.5%

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

   9. fma-def 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 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 9.2%

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

   1. diff-atan 18.8%

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

   2. associate--l+ 18.8%

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

   3. +-commutative 18.8%

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

   4. *-commutative 18.8%

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

   5. fma-def 18.8%

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

4. Applied egg-rr 18.8%

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

   1. +-commutative 18.8%

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

   2. associate-+l- 99.5%

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

   3. +-inverses 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in N around 0 99.5%

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

   1. unpow2 95.8%

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

   2. distribute-rgt1-in 95.8%

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

   3. +-commutative 95.8%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 96.2% 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 9.2%

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

   1. diff-atan 18.8%

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

   2. associate--l+ 18.8%

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

   3. +-commutative 18.8%

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

   4. *-commutative 18.8%

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

   5. fma-def 18.8%

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

4. Applied egg-rr 18.8%

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

   1. +-commutative 18.8%

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

   2. associate-+l- 99.5%

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

   3. +-inverses 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in N around inf 95.8%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + {N}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 95.8%

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

   2. distribute-rgt1-in 95.8%

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

   3. +-commutative 95.8%

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

9. Applied egg-rr 95.8%

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

10. Final simplification 95.8%

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

Alternative 4: 7.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

   1. diff-atan 18.8%

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

   2. associate--l+ 18.8%

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

   3. +-commutative 18.8%

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

   4. *-commutative 18.8%

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

   5. fma-def 18.8%

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

4. Applied egg-rr 18.8%

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

   1. +-commutative 18.8%

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

   2. associate-+l- 99.5%

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

   3. +-inverses 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in N around 0 8.0%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1 + N}} \]

8. Final simplification 8.0%

   \[\leadsto \tan^{-1}_* \frac{1}{1 + N} \]
9. Add Preprocessing

Alternative 5: 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 9.2%

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

   1. diff-atan 18.8%

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

   2. associate--l+ 18.8%

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

   3. +-commutative 18.8%

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

   4. *-commutative 18.8%

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

   5. fma-def 18.8%

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

4. Applied egg-rr 18.8%

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

   1. +-commutative 18.8%

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

   2. associate-+l- 99.5%

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

   3. +-inverses 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in N around 0 6.5%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1}} \]

8. Final simplification 6.5%

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

Alternative 6: 7.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

   1. diff-atan 18.8%

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

   2. associate--l+ 18.8%

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

   3. +-commutative 18.8%

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

   4. *-commutative 18.8%

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

   5. fma-def 18.8%

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

4. Applied egg-rr 18.8%

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

   1. +-commutative 18.8%

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

   2. associate-+l- 99.5%

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

   3. +-inverses 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in N around inf 95.8%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + {N}^{2}}} \]

8. Taylor expanded in N around 0 8.0%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N}} \]

9. Final simplification 8.0%

   \[\leadsto \tan^{-1}_* \frac{1}{N} \]
10. Add Preprocessing

Developer target: 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 2024041 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+100))

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

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