2atan (example 3.5)

Percentage Accurate: 76.6% → 99.4%

Time: 7.9s

Alternatives: 6

Speedup: 1.9×

Specification

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: 76.6% 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: 99.4% accurate, 1.0× 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]

Derivation

1. Initial program 74.1%

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

   1. diff-atan 75.6%

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

   2. associate--l+ 75.6%

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

   3. +-commutative 75.6%

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

   4. *-commutative 75.6%

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

   5. fma-def 75.6%

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

3. Applied egg-rr 75.6%

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

   1. +-commutative 75.6%

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

   2. associate-+l- 99.1%

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

   3. +-inverses 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -1 \lor \neg \left(N \leq 1\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -1.0) (not (<= N 1.0)))
   (atan2 1.0 (+ N (* N N)))
   (atan2 1.0 (+ 1.0 N))))

C

double code(double N) {
	double tmp;
	if ((N <= -1.0) || !(N <= 1.0)) {
		tmp = atan2(1.0, (N + (N * N)));
	} else {
		tmp = atan2(1.0, (1.0 + N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.0d0)) .or. (.not. (n <= 1.0d0))) then
        tmp = atan2(1.0d0, (n + (n * n)))
    else
        tmp = atan2(1.0d0, (1.0d0 + n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((N <= -1.0) || !(N <= 1.0)) {
		tmp = Math.atan2(1.0, (N + (N * N)));
	} else {
		tmp = Math.atan2(1.0, (1.0 + N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -1.0) or not (N <= 1.0):
		tmp = math.atan2(1.0, (N + (N * N)))
	else:
		tmp = math.atan2(1.0, (1.0 + N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -1.0) || !(N <= 1.0))
		tmp = atan(1.0, Float64(N + Float64(N * N)));
	else
		tmp = atan(1.0, Float64(1.0 + N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((N <= -1.0) || ~((N <= 1.0)))
		tmp = atan2(1.0, (N + (N * N)));
	else
		tmp = atan2(1.0, (1.0 + N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -1.0], N[Not[LessEqual[N, 1.0]], $MachinePrecision]], N[ArcTan[1.0 / N[(N + N[(N * N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(1.0 + N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -1 or 1 < N

   1. Initial program 49.9%

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

      1. diff-atan 52.7%

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

      2. associate--l+ 52.7%

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

      3. +-commutative 52.7%

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

      4. *-commutative 52.7%

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

      5. fma-def 52.7%

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

   3. Applied egg-rr 52.7%

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

      1. +-commutative 52.7%

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

      2. associate-+l- 98.4%

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

      3. +-inverses 98.4%

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

      4. metadata-eval 98.4%

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

      5. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in N around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. flip-+ 31.0%

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

      3. div-inv 31.0%

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

      4. fma-def 31.0%

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

      5. metadata-eval 31.0%

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

      6. pow-prod-up 31.0%

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

      7. metadata-eval 31.0%

         \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(1 - {N}^{\color{blue}{4}}, \frac{1}{1 - {N}^{2}}, N\right)} \]

      8. unpow2 31.0%

         \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(1 - {N}^{4}, \frac{1}{1 - \color{blue}{N \cdot N}}, N\right)} \]

   8. Applied egg-rr 31.0%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(1 - {N}^{4}, \frac{1}{1 - N \cdot N}, N\right)}} \]
   9. Step-by-step derivation

      1. fma-udef 31.0%

         \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(1 - {N}^{4}\right) \cdot \frac{1}{1 - N \cdot N} + N}} \]

      2. +-commutative 31.0%

         \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + \left(1 - {N}^{4}\right) \cdot \frac{1}{1 - N \cdot N}}} \]

      3. associate-*r/ 31.0%

         \[\leadsto \tan^{-1}_* \frac{1}{N + \color{blue}{\frac{\left(1 - {N}^{4}\right) \cdot 1}{1 - N \cdot N}}} \]

      4. *-rgt-identity 31.0%

         \[\leadsto \tan^{-1}_* \frac{1}{N + \frac{\color{blue}{1 - {N}^{4}}}{1 - N \cdot N}} \]

   10. Simplified 31.0%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + \frac{1 - {N}^{4}}{1 - N \cdot N}}} \]

   11. Taylor expanded in N around inf 96.2%

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

       1. unpow2 96.2%

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

   13. Simplified 96.2%

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

   if -1 < N < 1

   1. Initial program 100.0%

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

      1. diff-atan 99.9%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in N around 0 97.0%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -1 \lor \neg \left(N \leq 1\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \]

Alternative 3: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -1 \lor \neg \left(N \leq 1\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -1.0) (not (<= N 1.0))) (atan2 1.0 (* N N)) (atan2 1.0 1.0)))

C

double code(double N) {
	double tmp;
	if ((N <= -1.0) || !(N <= 1.0)) {
		tmp = atan2(1.0, (N * N));
	} else {
		tmp = atan2(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.0d0)) .or. (.not. (n <= 1.0d0))) then
        tmp = atan2(1.0d0, (n * n))
    else
        tmp = atan2(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((N <= -1.0) || !(N <= 1.0)) {
		tmp = Math.atan2(1.0, (N * N));
	} else {
		tmp = Math.atan2(1.0, 1.0);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -1.0) or not (N <= 1.0):
		tmp = math.atan2(1.0, (N * N))
	else:
		tmp = math.atan2(1.0, 1.0)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -1.0) || !(N <= 1.0))
		tmp = atan(1.0, Float64(N * N));
	else
		tmp = atan(1.0, 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((N <= -1.0) || ~((N <= 1.0)))
		tmp = atan2(1.0, (N * N));
	else
		tmp = atan2(1.0, 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -1.0], N[Not[LessEqual[N, 1.0]], $MachinePrecision]], N[ArcTan[1.0 / N[(N * N), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -1 or 1 < N

   1. Initial program 49.9%

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

      1. diff-atan 52.7%

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

      2. associate--l+ 52.7%

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

      3. +-commutative 52.7%

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

      4. *-commutative 52.7%

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

      5. fma-def 52.7%

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

   3. Applied egg-rr 52.7%

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

      1. +-commutative 52.7%

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

      2. associate-+l- 98.4%

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

      3. +-inverses 98.4%

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

      4. metadata-eval 98.4%

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

      5. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in N around inf 95.0%

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

      1. unpow2 95.0%

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

   8. Simplified 95.0%

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

   if -1 < N < 1

   1. Initial program 100.0%

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

      1. diff-atan 99.9%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in N around 0 95.4%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -1 \lor \neg \left(N \leq 1\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1}\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -0.62 \lor \neg \left(N \leq 1.65\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -0.62) (not (<= N 1.65)))
   (atan2 1.0 (* N N))
   (atan2 1.0 (+ 1.0 N))))

C

double code(double N) {
	double tmp;
	if ((N <= -0.62) || !(N <= 1.65)) {
		tmp = atan2(1.0, (N * N));
	} else {
		tmp = atan2(1.0, (1.0 + N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.62d0)) .or. (.not. (n <= 1.65d0))) then
        tmp = atan2(1.0d0, (n * n))
    else
        tmp = atan2(1.0d0, (1.0d0 + n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((N <= -0.62) || !(N <= 1.65)) {
		tmp = Math.atan2(1.0, (N * N));
	} else {
		tmp = Math.atan2(1.0, (1.0 + N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -0.62) or not (N <= 1.65):
		tmp = math.atan2(1.0, (N * N))
	else:
		tmp = math.atan2(1.0, (1.0 + N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -0.62) || !(N <= 1.65))
		tmp = atan(1.0, Float64(N * N));
	else
		tmp = atan(1.0, Float64(1.0 + N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((N <= -0.62) || ~((N <= 1.65)))
		tmp = atan2(1.0, (N * N));
	else
		tmp = atan2(1.0, (1.0 + N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -0.62], N[Not[LessEqual[N, 1.65]], $MachinePrecision]], N[ArcTan[1.0 / N[(N * N), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(1.0 + N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -0.619999999999999996 or 1.6499999999999999 < N

   1. Initial program 49.9%

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

      1. diff-atan 52.7%

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

      2. associate--l+ 52.7%

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

      3. +-commutative 52.7%

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

      4. *-commutative 52.7%

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

      5. fma-def 52.7%

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

   3. Applied egg-rr 52.7%

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

      1. +-commutative 52.7%

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

      2. associate-+l- 98.4%

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

      3. +-inverses 98.4%

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

      4. metadata-eval 98.4%

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

      5. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in N around inf 95.0%

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

      1. unpow2 95.0%

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

   8. Simplified 95.0%

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

   if -0.619999999999999996 < N < 1.6499999999999999

   1. Initial program 100.0%

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

      1. diff-atan 99.9%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in N around 0 97.0%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -0.62 \lor \neg \left(N \leq 1.65\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. diff-atan 75.6%

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

   2. associate--l+ 75.6%

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

   3. +-commutative 75.6%

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

   4. *-commutative 75.6%

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

   5. fma-def 75.6%

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

3. Applied egg-rr 75.6%

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

   1. +-commutative 75.6%

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

   2. associate-+l- 99.1%

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

   3. +-inverses 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in N around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. flip-+ 64.4%

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

   3. div-inv 64.4%

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

   4. fma-def 64.4%

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

   5. metadata-eval 64.4%

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

   6. pow-prod-up 64.4%

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

   7. metadata-eval 64.4%

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(1 - {N}^{\color{blue}{4}}, \frac{1}{1 - {N}^{2}}, N\right)} \]

   8. unpow2 64.4%

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(1 - {N}^{4}, \frac{1}{1 - \color{blue}{N \cdot N}}, N\right)} \]

8. Applied egg-rr 64.4%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(1 - {N}^{4}, \frac{1}{1 - N \cdot N}, N\right)}} \]
9. Step-by-step derivation

   1. fma-udef 64.4%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(1 - {N}^{4}\right) \cdot \frac{1}{1 - N \cdot N} + N}} \]

   2. +-commutative 64.4%

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + \left(1 - {N}^{4}\right) \cdot \frac{1}{1 - N \cdot N}}} \]

   3. associate-*r/ 64.4%

      \[\leadsto \tan^{-1}_* \frac{1}{N + \color{blue}{\frac{\left(1 - {N}^{4}\right) \cdot 1}{1 - N \cdot N}}} \]

   4. *-rgt-identity 64.4%

      \[\leadsto \tan^{-1}_* \frac{1}{N + \frac{\color{blue}{1 - {N}^{4}}}{1 - N \cdot N}} \]

10. Simplified 64.4%

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N + \frac{1 - {N}^{4}}{1 - N \cdot N}}} \]
11. Step-by-step derivation

    1. metadata-eval 64.4%

       \[\leadsto \tan^{-1}_* \frac{1}{N + \frac{\color{blue}{1 \cdot 1} - {N}^{4}}{1 - N \cdot N}} \]

    2. metadata-eval 64.4%

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

    3. pow-prod-up 64.4%

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

    4. pow-prod-down 64.4%

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

    5. pow2 64.4%

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

    6. flip-+ 99.1%

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

    7. +-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

13. Final simplification 99.1%

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

Alternative 6: 51.0% 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 74.1%

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

   1. diff-atan 75.6%

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

   2. associate--l+ 75.6%

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

   3. +-commutative 75.6%

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

   4. *-commutative 75.6%

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

   5. fma-def 75.6%

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

3. Applied egg-rr 75.6%

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

   1. +-commutative 75.6%

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

   2. associate-+l- 99.1%

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

   3. +-inverses 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in N around 0 48.6%

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

7. Final simplification 48.6%

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

Developer target: 99.4% 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 2023278 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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