2atan (example 3.5)

Percentage Accurate: 76.5% → 99.5%

Time: 7.6s

Alternatives: 7

Speedup: 1.8×

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: 76.5% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. diff-atan 72.9%

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

   2. +-commutative 72.9%

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

   3. associate--l+ 99.7%

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

   4. +-inverses 99.7%

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

   5. metadata-eval 99.7%

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

   6. distribute-lft1-in 99.7%

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

   7. +-commutative 99.7%

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

   8. pow2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. +-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. fma-def 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \mathbf{if}\;t_0 \leq 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot \left(1 + N\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (atan (+ 1.0 N)) (atan N))))
   (if (<= t_0 1e-8) (atan2 1.0 (* N (+ 1.0 N))) t_0)))

C

double code(double N) {
	double t_0 = atan((1.0 + N)) - atan(N);
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = atan2(1.0, (N * (1.0 + N)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan((1.0d0 + n)) - atan(n)
    if (t_0 <= 1d-8) then
        tmp = atan2(1.0d0, (n * (1.0d0 + n)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double t_0 = Math.atan((1.0 + N)) - Math.atan(N);
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = Math.atan2(1.0, (N * (1.0 + N)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(N):
	t_0 = math.atan((1.0 + N)) - math.atan(N)
	tmp = 0
	if t_0 <= 1e-8:
		tmp = math.atan2(1.0, (N * (1.0 + N)))
	else:
		tmp = t_0
	return tmp

Julia

function code(N)
	t_0 = Float64(atan(Float64(1.0 + N)) - atan(N))
	tmp = 0.0
	if (t_0 <= 1e-8)
		tmp = atan(1.0, Float64(N * Float64(1.0 + N)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	t_0 = atan((1.0 + N)) - atan(N);
	tmp = 0.0;
	if (t_0 <= 1e-8)
		tmp = atan2(1.0, (N * (1.0 + N)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := Block[{t$95$0 = N[(N[ArcTan[N[(1.0 + N), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-8], N[ArcTan[1.0 / N[(N * N[(1.0 + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (atan.f64 (+.f64 N 1)) (atan.f64 N)) < 1e-8

   1. Initial program 48.9%

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

      1. diff-atan 51.9%

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

      2. +-commutative 51.9%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft1-in 99.4%

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

      7. +-commutative 99.4%

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

      8. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-+r+ 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in N around inf 98.9%

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

      1. unpow2 98.9%

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

      2. distribute-rgt1-in 98.9%

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

      3. +-commutative 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 1e-8 < (-.f64 (atan.f64 (+.f64 N 1)) (atan.f64 N))

   1. Initial program 99.8%

      \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N \leq 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot \left(1 + N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \mathbf{if}\;t_0 \leq 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + {N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (atan (+ 1.0 N)) (atan N))))
   (if (<= t_0 1e-8) (atan2 1.0 (+ N (pow N 2.0))) t_0)))

C

double code(double N) {
	double t_0 = atan((1.0 + N)) - atan(N);
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = atan2(1.0, (N + pow(N, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan((1.0d0 + n)) - atan(n)
    if (t_0 <= 1d-8) then
        tmp = atan2(1.0d0, (n + (n ** 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double t_0 = Math.atan((1.0 + N)) - Math.atan(N);
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = Math.atan2(1.0, (N + Math.pow(N, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(N):
	t_0 = math.atan((1.0 + N)) - math.atan(N)
	tmp = 0
	if t_0 <= 1e-8:
		tmp = math.atan2(1.0, (N + math.pow(N, 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(N)
	t_0 = Float64(atan(Float64(1.0 + N)) - atan(N))
	tmp = 0.0
	if (t_0 <= 1e-8)
		tmp = atan(1.0, Float64(N + (N ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	t_0 = atan((1.0 + N)) - atan(N);
	tmp = 0.0;
	if (t_0 <= 1e-8)
		tmp = atan2(1.0, (N + (N ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := Block[{t$95$0 = N[(N[ArcTan[N[(1.0 + N), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-8], N[ArcTan[1.0 / N[(N + N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (atan.f64 (+.f64 N 1)) (atan.f64 N)) < 1e-8

   1. Initial program 48.9%

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

      1. diff-atan 51.9%

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

      2. +-commutative 51.9%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft1-in 99.4%

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

      7. +-commutative 99.4%

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

      8. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-+r+ 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in N around inf 98.9%

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

   if 1e-8 < (-.f64 (atan.f64 (+.f64 N 1)) (atan.f64 N))

   1. Initial program 99.8%

      \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N \leq 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + {N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -0.56 \lor \neg \left(N \leq 1.05\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot \left(1 + N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{\frac{1}{1 - N \cdot \left(N + -1\right)} \cdot \left(1 + N \cdot \left(N + 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -0.56) (not (<= N 1.05)))
   (atan2 1.0 (* N (+ 1.0 N)))
   (atan2 1.0 (* (/ 1.0 (- 1.0 (* N (+ N -1.0)))) (+ 1.0 (* N (+ N 2.0)))))))

C

double code(double N) {
	double tmp;
	if ((N <= -0.56) || !(N <= 1.05)) {
		tmp = atan2(1.0, (N * (1.0 + N)));
	} else {
		tmp = atan2(1.0, ((1.0 / (1.0 - (N * (N + -1.0)))) * (1.0 + (N * (N + 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.56d0)) .or. (.not. (n <= 1.05d0))) then
        tmp = atan2(1.0d0, (n * (1.0d0 + n)))
    else
        tmp = atan2(1.0d0, ((1.0d0 / (1.0d0 - (n * (n + (-1.0d0))))) * (1.0d0 + (n * (n + 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((N <= -0.56) || !(N <= 1.05)) {
		tmp = Math.atan2(1.0, (N * (1.0 + N)));
	} else {
		tmp = Math.atan2(1.0, ((1.0 / (1.0 - (N * (N + -1.0)))) * (1.0 + (N * (N + 2.0)))));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -0.56) or not (N <= 1.05):
		tmp = math.atan2(1.0, (N * (1.0 + N)))
	else:
		tmp = math.atan2(1.0, ((1.0 / (1.0 - (N * (N + -1.0)))) * (1.0 + (N * (N + 2.0)))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -0.56) || !(N <= 1.05))
		tmp = atan(1.0, Float64(N * Float64(1.0 + N)));
	else
		tmp = atan(1.0, Float64(Float64(1.0 / Float64(1.0 - Float64(N * Float64(N + -1.0)))) * Float64(1.0 + Float64(N * Float64(N + 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((N <= -0.56) || ~((N <= 1.05)))
		tmp = atan2(1.0, (N * (1.0 + N)));
	else
		tmp = atan2(1.0, ((1.0 / (1.0 - (N * (N + -1.0)))) * (1.0 + (N * (N + 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -0.56], N[Not[LessEqual[N, 1.05]], $MachinePrecision]], N[ArcTan[1.0 / N[(N * N[(1.0 + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(N[(1.0 / N[(1.0 - N[(N * N[(N + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N * N[(N + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -0.56000000000000005 or 1.05000000000000004 < N

   1. Initial program 49.2%

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

      1. diff-atan 52.2%

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

      2. +-commutative 52.2%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft1-in 99.4%

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

      7. +-commutative 99.4%

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

      8. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-+r+ 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in N around inf 98.5%

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

      1. unpow2 98.5%

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

      2. distribute-rgt1-in 98.5%

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

      3. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   if -0.56000000000000005 < N < 1.05000000000000004

   1. Initial program 100.0%

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

      1. diff-atan 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. +-commutative 100.0%

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

      8. pow2 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+r+ 100.0%

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

   5. Simplified 100.0%

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

      1. flip-+ 100.0%

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

      2. div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. metadata-eval 99.9%

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

      7. unpow2 99.9%

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

      8. +-commutative 99.9%

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

      9. metadata-eval 99.9%

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

      10. unpow2 99.9%

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

      11. associate--r- 99.9%

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

      12. unpow2 99.9%

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

      13. *-un-lft-identity 99.9%

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

      14. distribute-rgt-out-- 99.9%

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

      15. pow2 99.9%

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

      16. pow-sqr 99.9%

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

      17. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in N around 0 100.0%

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

      1. unpow2 100.0%

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

      2. distribute-rgt-out 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -0.56 \lor \neg \left(N \leq 1.05\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot \left(1 + N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{\frac{1}{1 - N \cdot \left(N + -1\right)} \cdot \left(1 + N \cdot \left(N + 2\right)\right)}\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -1.0) (not (<= N 1.0)))
   (atan2 1.0 (* N (+ 1.0 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 * (1.0 + 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 * (1.0d0 + 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 * (1.0 + 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 * (1.0 + 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(1.0 + 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 * (1.0 + 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[(1.0 + 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 49.2%

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

      1. diff-atan 52.2%

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

      2. +-commutative 52.2%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft1-in 99.4%

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

      7. +-commutative 99.4%

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

      8. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-+r+ 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in N around inf 98.5%

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

      1. unpow2 98.5%

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

      2. distribute-rgt1-in 98.5%

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

      3. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   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 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. +-commutative 100.0%

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

      8. pow2 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in N around 0 99.2%

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

4. Final simplification 98.8%

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

Alternative 6: 51.7% 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 71.2%

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

   1. diff-atan 72.9%

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

   2. +-commutative 72.9%

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

   3. associate--l+ 99.7%

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

   4. +-inverses 99.7%

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

   5. metadata-eval 99.7%

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

   6. distribute-lft1-in 99.7%

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

   7. +-commutative 99.7%

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

   8. pow2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-+r+ 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in N around 0 46.2%

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

7. Final simplification 46.2%

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

Alternative 7: 50.5% 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 71.2%

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

   1. diff-atan 72.9%

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

   2. +-commutative 72.9%

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

   3. associate--l+ 99.7%

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

   4. +-inverses 99.7%

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

   5. metadata-eval 99.7%

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

   6. distribute-lft1-in 99.7%

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

   7. +-commutative 99.7%

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

   8. pow2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-+r+ 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in N around 0 44.8%

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

7. Final simplification 44.8%

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

Developer target: 99.5% 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 2023336 
(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)))