2atan (example 3.5)

Percentage Accurate: 9.0% → 99.6%

Time: 9.9s

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\tan^{-1} N, \tan^{-1}_* \frac{N + \left(N + 1\right)}{1 - \mathsf{fma}\left(N, N, N\right)}, {\tan^{-1} \left(N + 1\right)}^{2}\right)\\ t\_0 \cdot \frac{\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0
         (fma
          (atan N)
          (atan2 (+ N (+ N 1.0)) (- 1.0 (fma N N N)))
          (pow (atan (+ N 1.0)) 2.0))))
   (* t_0 (/ (atan2 1.0 (fma N N (+ N 1.0))) t_0))))

C

double code(double N) {
	double t_0 = fma(atan(N), atan2((N + (N + 1.0)), (1.0 - fma(N, N, N))), pow(atan((N + 1.0)), 2.0));
	return t_0 * (atan2(1.0, fma(N, N, (N + 1.0))) / t_0);
}

Julia

function code(N)
	t_0 = fma(atan(N), atan(Float64(N + Float64(N + 1.0)), Float64(1.0 - fma(N, N, N))), (atan(Float64(N + 1.0)) ^ 2.0))
	return Float64(t_0 * Float64(atan(1.0, fma(N, N, Float64(N + 1.0))) / t_0))
end

Wolfram

code[N_] := Block[{t$95$0 = N[(N[ArcTan[N], $MachinePrecision] * N[ArcTan[N[(N + N[(N + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N * N + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(N[ArcTan[1.0 / N[(N * N + N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 9.5%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing

4. Applied egg-rr 99.5%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	return atan(1.0, fma(N, N, Float64(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 9.5%

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

   1. lift-+.f64 N/A

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

   2. diff-atan N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-atan2.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 3: 96.2% 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 9.5%

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

   1. lift-+.f64 N/A

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

   2. diff-atan N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-atan2.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. rgt-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

   6. *-rgt-identity N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 95.6

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

7. Simplified 95.6%

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

Alternative 4: 93.1% 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 9.5%

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

   1. lift-+.f64 N/A

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

   2. diff-atan N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-atan2.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

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 92.2

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

7. Simplified 92.2%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot N}} \]
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 9.5%

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

   1. lift-+.f64 N/A

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

   2. diff-atan N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-atan2.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

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. lower-+.f64 8.0

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

7. Simplified 8.0%

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

Alternative 6: 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 9.5%

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

   1. lift-+.f64 N/A

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

   2. diff-atan N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower-atan2.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in N around 0

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

7. Simplified 6.5%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1}} \]
8. 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 2024215 
(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)))