2atan (example 3.5)

Percentage Accurate: 75.4% → 99.4%

Time: 5.8s

Alternatives: 7

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: 75.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. diff-atan 78.7%

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

   2. associate--l+ 78.7%

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

   3. +-commutative 78.7%

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

   4. *-commutative 78.7%

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

   5. fma-def 78.7%

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

3. Applied egg-rr 78.7%

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

   1. +-commutative 78.7%

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

   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. Final simplification 99.9%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (atan (+ 1.0 N)) (atan N))))
   (if (<= t_0 5e-9) (atan2 1.0 (* (- -0.5 N) (- -0.5 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, ((-0.5 - N) * (-0.5 - 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, (((-0.5d0) - n) * ((-0.5d0) - 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, ((-0.5 - N) * (-0.5 - 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, ((-0.5 - N) * (-0.5 - 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(Float64(-0.5 - N) * Float64(-0.5 - 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, ((-0.5 - N) * (-0.5 - 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[(-0.5 - N), $MachinePrecision] * N[(-0.5 - 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 54.3%

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

      1. diff-atan 59.0%

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

      2. associate--l+ 59.0%

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

      3. +-commutative 59.0%

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

      4. *-commutative 59.0%

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

      5. fma-def 59.0%

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

   3. Applied egg-rr 59.0%

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

      1. +-commutative 59.0%

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

      2. associate-+l- 99.8%

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

      3. +-inverses 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in N around -inf 99.3%

      \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N - 0.5\right)}}^{2}} \]
   9. Step-by-step derivation

      1. sub-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N + \left(-0.5\right)\right)}}^{2}} \]

      2. +-commutative 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) + -1 \cdot N\right)}}^{2}} \]

      3. mul-1-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\left(-0.5\right) + \color{blue}{\left(-N\right)}\right)}^{2}} \]

      4. unsub-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) - N\right)}}^{2}} \]

      5. metadata-eval 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\color{blue}{-0.5} - N\right)}^{2}} \]

   10. Simplified 99.3%

       \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-0.5 - N\right)}}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 99.3%

          \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

   12. Applied egg-rr 99.3%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

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

   1. Initial program 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -0.86:\\ \;\;\;\;\tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}\\ \mathbf{elif}\;N \leq 0.86:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + 1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{-0.5 \cdot \left(-0.5 - N\right) + N \cdot \left(N - -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N -0.86)
   (atan2 1.0 (* (- -0.5 N) (- -0.5 N)))
   (if (<= N 0.86)
     (atan2 1.0 (+ N 1.0))
     (atan2 1.0 (+ (* -0.5 (- -0.5 N)) (* N (- N -0.5)))))))

C

double code(double N) {
	double tmp;
	if (N <= -0.86) {
		tmp = atan2(1.0, ((-0.5 - N) * (-0.5 - N)));
	} else if (N <= 0.86) {
		tmp = atan2(1.0, (N + 1.0));
	} else {
		tmp = atan2(1.0, ((-0.5 * (-0.5 - N)) + (N * (N - -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-0.86d0)) then
        tmp = atan2(1.0d0, (((-0.5d0) - n) * ((-0.5d0) - n)))
    else if (n <= 0.86d0) then
        tmp = atan2(1.0d0, (n + 1.0d0))
    else
        tmp = atan2(1.0d0, (((-0.5d0) * ((-0.5d0) - n)) + (n * (n - (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= -0.86) {
		tmp = Math.atan2(1.0, ((-0.5 - N) * (-0.5 - N)));
	} else if (N <= 0.86) {
		tmp = Math.atan2(1.0, (N + 1.0));
	} else {
		tmp = Math.atan2(1.0, ((-0.5 * (-0.5 - N)) + (N * (N - -0.5))));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= -0.86:
		tmp = math.atan2(1.0, ((-0.5 - N) * (-0.5 - N)))
	elif N <= 0.86:
		tmp = math.atan2(1.0, (N + 1.0))
	else:
		tmp = math.atan2(1.0, ((-0.5 * (-0.5 - N)) + (N * (N - -0.5))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= -0.86)
		tmp = atan(1.0, Float64(Float64(-0.5 - N) * Float64(-0.5 - N)));
	elseif (N <= 0.86)
		tmp = atan(1.0, Float64(N + 1.0));
	else
		tmp = atan(1.0, Float64(Float64(-0.5 * Float64(-0.5 - N)) + Float64(N * Float64(N - -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= -0.86)
		tmp = atan2(1.0, ((-0.5 - N) * (-0.5 - N)));
	elseif (N <= 0.86)
		tmp = atan2(1.0, (N + 1.0));
	else
		tmp = atan2(1.0, ((-0.5 * (-0.5 - N)) + (N * (N - -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, -0.86], N[ArcTan[1.0 / N[(N[(-0.5 - N), $MachinePrecision] * N[(-0.5 - N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N, 0.86], N[ArcTan[1.0 / N[(N + 1.0), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(N[(-0.5 * N[(-0.5 - N), $MachinePrecision]), $MachinePrecision] + N[(N * N[(N - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if N < -0.859999999999999987

   1. Initial program 60.3%

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

      1. diff-atan 63.8%

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

      2. associate--l+ 63.8%

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

      3. +-commutative 63.8%

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

      4. *-commutative 63.8%

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

      5. fma-def 63.8%

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

   3. Applied egg-rr 63.8%

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

      1. +-commutative 63.8%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 99.8%

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

      2. pow2 99.8%

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

      3. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in N around -inf 99.8%

      \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N - 0.5\right)}}^{2}} \]
   9. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N + \left(-0.5\right)\right)}}^{2}} \]

      2. +-commutative 99.8%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) + -1 \cdot N\right)}}^{2}} \]

      3. mul-1-neg 99.8%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\left(-0.5\right) + \color{blue}{\left(-N\right)}\right)}^{2}} \]

      4. unsub-neg 99.8%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) - N\right)}}^{2}} \]

      5. metadata-eval 99.8%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\color{blue}{-0.5} - N\right)}^{2}} \]

   10. Simplified 99.8%

       \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-0.5 - N\right)}}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 99.8%

          \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

   if -0.859999999999999987 < N < 0.859999999999999987

   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. 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- 100.0%

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

      3. +-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in N around 0 99.2%

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

   if 0.859999999999999987 < N

   1. Initial program 47.6%

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

      1. diff-atan 53.6%

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

      2. associate--l+ 53.6%

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

      3. +-commutative 53.6%

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

      4. *-commutative 53.6%

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

      5. fma-def 53.6%

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

   3. Applied egg-rr 53.6%

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

      1. +-commutative 53.6%

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

      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%

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

   5. Simplified 99.6%

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

      1. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in N around -inf 98.7%

      \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N - 0.5\right)}}^{2}} \]
   9. Step-by-step derivation

      1. sub-neg 98.7%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N + \left(-0.5\right)\right)}}^{2}} \]

      2. +-commutative 98.7%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) + -1 \cdot N\right)}}^{2}} \]

      3. mul-1-neg 98.7%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\left(-0.5\right) + \color{blue}{\left(-N\right)}\right)}^{2}} \]

      4. unsub-neg 98.7%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) - N\right)}}^{2}} \]

      5. metadata-eval 98.7%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\color{blue}{-0.5} - N\right)}^{2}} \]

   10. Simplified 98.7%

       \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-0.5 - N\right)}}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 98.7%

          \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

       2. sub-neg 98.7%

          \[\leadsto \tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \color{blue}{\left(-0.5 + \left(-N\right)\right)}} \]

       3. distribute-lft-in 98.7%

          \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot -0.5 + \left(-0.5 - N\right) \cdot \left(-N\right)}} \]

   12. Applied egg-rr 98.7%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot -0.5 + \left(-0.5 - N\right) \cdot \left(-N\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -0.86:\\ \;\;\;\;\tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}\\ \mathbf{elif}\;N \leq 0.86:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + 1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{-0.5 \cdot \left(-0.5 - N\right) + N \cdot \left(N - -0.5\right)}\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -0.86 \lor \neg \left(N \leq 0.86\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -0.86) (not (<= N 0.86)))
   (atan2 1.0 (* (- -0.5 N) (- -0.5 N)))
   (atan2 1.0 (+ N 1.0))))

C

double code(double N) {
	double tmp;
	if ((N <= -0.86) || !(N <= 0.86)) {
		tmp = atan2(1.0, ((-0.5 - N) * (-0.5 - N)));
	} else {
		tmp = atan2(1.0, (N + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.86d0)) .or. (.not. (n <= 0.86d0))) then
        tmp = atan2(1.0d0, (((-0.5d0) - n) * ((-0.5d0) - n)))
    else
        tmp = atan2(1.0d0, (n + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((N <= -0.86) || !(N <= 0.86)) {
		tmp = Math.atan2(1.0, ((-0.5 - N) * (-0.5 - N)));
	} else {
		tmp = Math.atan2(1.0, (N + 1.0));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -0.86) or not (N <= 0.86):
		tmp = math.atan2(1.0, ((-0.5 - N) * (-0.5 - N)))
	else:
		tmp = math.atan2(1.0, (N + 1.0))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -0.86) || !(N <= 0.86))
		tmp = atan(1.0, Float64(Float64(-0.5 - N) * Float64(-0.5 - N)));
	else
		tmp = atan(1.0, Float64(N + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((N <= -0.86) || ~((N <= 0.86)))
		tmp = atan2(1.0, ((-0.5 - N) * (-0.5 - N)));
	else
		tmp = atan2(1.0, (N + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -0.86], N[Not[LessEqual[N, 0.86]], $MachinePrecision]], N[ArcTan[1.0 / N[(N[(-0.5 - N), $MachinePrecision] * N[(-0.5 - N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(N + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -0.859999999999999987 or 0.859999999999999987 < N

   1. Initial program 54.3%

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

      1. diff-atan 59.0%

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

      2. associate--l+ 59.0%

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

      3. +-commutative 59.0%

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

      4. *-commutative 59.0%

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

      5. fma-def 59.0%

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

   3. Applied egg-rr 59.0%

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

      1. +-commutative 59.0%

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

      2. associate-+l- 99.8%

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

      3. +-inverses 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in N around -inf 99.3%

      \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N - 0.5\right)}}^{2}} \]
   9. Step-by-step derivation

      1. sub-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-1 \cdot N + \left(-0.5\right)\right)}}^{2}} \]

      2. +-commutative 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) + -1 \cdot N\right)}}^{2}} \]

      3. mul-1-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\left(-0.5\right) + \color{blue}{\left(-N\right)}\right)}^{2}} \]

      4. unsub-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(\left(-0.5\right) - N\right)}}^{2}} \]

      5. metadata-eval 99.3%

         \[\leadsto \tan^{-1}_* \frac{1}{{\left(\color{blue}{-0.5} - N\right)}^{2}} \]

   10. Simplified 99.3%

       \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{\left(-0.5 - N\right)}}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 99.3%

          \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

   12. Applied egg-rr 99.3%

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}} \]

   if -0.859999999999999987 < N < 0.859999999999999987

   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. 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- 100.0%

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

      3. +-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -0.86 \lor \neg \left(N \leq 0.86\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{\left(-0.5 - N\right) \cdot \left(-0.5 - N\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + 1}\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double N) {
	double tmp;
	if ((N <= -1.0) || !(N <= 1.0)) {
		tmp = atan2(1.0, (N * (N + 1.0)));
	} else {
		tmp = atan2(1.0, (N + 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 + 1.0d0)))
    else
        tmp = atan2(1.0d0, (n + 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 + 1.0)));
	} else {
		tmp = Math.atan2(1.0, (N + 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 + 1.0)))
	else:
		tmp = math.atan2(1.0, (N + 1.0))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -1.0) || !(N <= 1.0))
		tmp = atan(1.0, Float64(N * Float64(N + 1.0)));
	else
		tmp = atan(1.0, Float64(N + 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 + 1.0)));
	else
		tmp = atan2(1.0, (N + 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[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(N + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -1 or 1 < N

   1. Initial program 54.3%

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

      1. diff-atan 59.0%

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

      2. associate--l+ 59.0%

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

      3. +-commutative 59.0%

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

      4. *-commutative 59.0%

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

      5. fma-def 59.0%

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

   3. Applied egg-rr 59.0%

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

      1. +-commutative 59.0%

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

      2. associate-+l- 99.8%

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

      3. +-inverses 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in N around inf 99.3%

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

      1. unpow2 99.3%

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

      2. distribute-rgt1-in 99.3%

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

   8. Applied egg-rr 99.3%

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

   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. 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- 100.0%

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

      3. +-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   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 99.2%

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

Alternative 6: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. diff-atan 78.7%

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

   2. associate--l+ 78.7%

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

   3. +-commutative 78.7%

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

   4. *-commutative 78.7%

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

   5. fma-def 78.7%

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

3. Applied egg-rr 78.7%

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

   1. +-commutative 78.7%

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

   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 50.6%

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

7. Final simplification 50.6%

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

Alternative 7: 49.7% 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 76.3%

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

   1. diff-atan 78.7%

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

   2. associate--l+ 78.7%

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

   3. +-commutative 78.7%

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

   4. *-commutative 78.7%

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

   5. fma-def 78.7%

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

3. Applied egg-rr 78.7%

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

   1. +-commutative 78.7%

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

   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 49.4%

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

7. Final simplification 49.4%

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

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 2023311 
(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)))