2atan (example 3.5)

Percentage Accurate: 8.6% → 99.6%

Time: 5.7s

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: 8.6% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.4%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
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. lift-atan.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. diff-atan N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower-atan2.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. associate-+r+ N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-+.f64 N/A

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

   24. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. lift-fma.f64 N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.4%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
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. lift-atan.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. diff-atan N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower-atan2.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. associate-+r+ N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-+.f64 N/A

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

   24. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 3: 96.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.4%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
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. lift-atan.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. diff-atan N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower-atan2.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. associate-+r+ N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-+.f64 N/A

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

   24. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in N around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. unpow2 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. rgt-mult-inverse N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 95.8

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

7. Applied rewrites 95.8%

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

Alternative 4: 93.3% accurate, 1.9× 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 9.4%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
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. lift-atan.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. diff-atan N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower-atan2.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. associate-+r+ N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-+.f64 N/A

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

   24. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in N around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 93.2

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

7. Applied rewrites 93.2%

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

Alternative 5: 7.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.4%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
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. lift-atan.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. diff-atan N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower-atan2.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. associate-+r+ N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-+.f64 N/A

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

   24. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in N around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. lower--.f64 7.9

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

7. Applied rewrites 7.9%

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

Alternative 6: 6.3% 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 9.4%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
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. lift-atan.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. diff-atan N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower-atan2.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. associate-+r+ N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-+.f64 N/A

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

   24. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in N around 0

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

7. Applied rewrites 6.3%

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

Developer Target 1: 99.6% accurate, 1.9× speedup

Math

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

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 2024313 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+100))

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

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