2atan (example 3.5)

Percentage Accurate: 8.6% → 99.6%

Time: 5.1s

Alternatives: 7

Speedup: 2.0×

Specification

\[N > 1 \land N < 10^{+100}\]

Math

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

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

Julia

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

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

Wolfram

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

Initial Program: 8.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

Julia

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

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

Wolfram

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

Alternative 1: 99.6% accurate, 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 10.6%

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

   1. diff-atan 21.7%

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

   2. associate--l+ 21.8%

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

   3. +-commutative 21.8%

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

   4. *-commutative 21.8%

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

   5. fma-def 21.8%

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

4. Applied egg-rr 21.8%

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

   1. +-commutative 21.8%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. +-commutative 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. *-un-lft-identity 99.7%

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

   5. fma-def 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot \left(1 + N\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (atan (+ 1.0 N)) (atan N))))
   (if (<= t_0 5e-9) (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 <= 5e-9) {
		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 <= 5d-9) 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 <= 5e-9) {
		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 <= 5e-9:
		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 <= 5e-9)
		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 <= 5e-9)
		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, 5e-9], 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)) < 5.0000000000000001e-9

   1. Initial program 6.1%

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

      1. diff-atan 16.9%

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

      2. associate--l+ 16.9%

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

      3. +-commutative 16.9%

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

      4. *-commutative 16.9%

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

      5. fma-def 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. +-commutative 16.9%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in N around inf 98.6%

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

      1. unpow2 98.6%

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

      2. distribute-rgt1-in 98.6%

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

      3. +-commutative 98.6%

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

   9. Applied egg-rr 98.6%

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

   if 5.0000000000000001e-9 < (-.f64 (atan.f64 (+.f64 N 1)) (atan.f64 N))

   1. Initial program 83.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot \left(1 + N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + {N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (atan (+ 1.0 N)) (atan N))))
   (if (<= t_0 5e-9) (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 <= 5e-9) {
		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 <= 5d-9) 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 <= 5e-9) {
		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 <= 5e-9:
		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 <= 5e-9)
		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 <= 5e-9)
		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, 5e-9], 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)) < 5.0000000000000001e-9

   1. Initial program 6.1%

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

      1. diff-atan 16.9%

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

      2. associate--l+ 16.9%

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

      3. +-commutative 16.9%

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

      4. *-commutative 16.9%

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

      5. fma-def 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. +-commutative 16.9%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in N around inf 98.6%

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

   if 5.0000000000000001e-9 < (-.f64 (atan.f64 (+.f64 N 1)) (atan.f64 N))

   1. Initial program 83.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + {N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.6%

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

   1. diff-atan 21.7%

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

   2. associate--l+ 21.8%

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

   3. +-commutative 21.8%

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

   4. *-commutative 21.8%

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

   5. fma-def 21.8%

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

4. Applied egg-rr 21.8%

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

   1. +-commutative 21.8%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in N around inf 95.1%

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

   1. unpow2 95.1%

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

   2. distribute-rgt1-in 95.1%

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

   3. +-commutative 95.1%

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

9. Applied egg-rr 95.1%

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

10. Final simplification 95.1%

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

Alternative 5: 93.3% accurate, 2.0× 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 10.6%

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

   1. diff-atan 21.7%

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

   2. associate--l+ 21.8%

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

   3. +-commutative 21.8%

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

   4. *-commutative 21.8%

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

   5. fma-def 21.8%

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

4. Applied egg-rr 21.8%

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

   1. +-commutative 21.8%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. add-cbrt-cube 54.4%

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

   2. pow3 54.5%

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

8. Applied egg-rr 54.5%

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

9. Taylor expanded in N around inf 46.3%

   \[\leadsto \tan^{-1}_* \frac{1}{\sqrt[3]{\color{blue}{{N}^{6}}}} \]
10. Step-by-step derivation

    1. pow1/3 43.5%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{\left({N}^{6}\right)}^{0.3333333333333333}}} \]

    2. pow-pow 91.5%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{\left(6 \cdot 0.3333333333333333\right)}}} \]

    3. metadata-eval 91.5%

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

    4. unpow2 91.5%

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

11. Applied egg-rr 91.5%

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

12. Final simplification 91.5%

    \[\leadsto \tan^{-1}_* \frac{1}{N \cdot N} \]
13. 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 10.6%

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

   1. diff-atan 21.7%

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

   2. associate--l+ 21.8%

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

   3. +-commutative 21.8%

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

   4. *-commutative 21.8%

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

   5. fma-def 21.8%

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

4. Applied egg-rr 21.8%

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

   1. +-commutative 21.8%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in N around 0 6.5%

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

8. Final simplification 6.5%

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

Alternative 7: 7.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.6%

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

   1. diff-atan 21.7%

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

   2. associate--l+ 21.8%

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

   3. +-commutative 21.8%

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

   4. *-commutative 21.8%

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

   5. fma-def 21.8%

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

4. Applied egg-rr 21.8%

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

   1. +-commutative 21.8%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in N around inf 95.1%

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

8. Taylor expanded in N around 0 8.0%

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

9. Final simplification 8.0%

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

Developer target: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0))))))

C

double code(double N) {
	return atan((1.0 / (1.0 + (N * (N + 1.0)))));
}

Fortran

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

Java

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

Python

def code(N):
	return math.atan((1.0 / (1.0 + (N * (N + 1.0)))))

Julia

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

MATLAB

function tmp = code(N)
	tmp = atan((1.0 / (1.0 + (N * (N + 1.0)))));
end

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+100))

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

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