2atan (example 3.5)

Percentage Accurate: 76.0% → 99.4%

Time: 7.8s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. diff-atan 78.8%

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

   2. +-commutative 78.8%

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

   3. associate--l+ 99.2%

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

   4. +-inverses 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. associate-+r+ 99.2%

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

   8. +-commutative 99.2%

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

   9. +-commutative 99.2%

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

   10. fma-def 99.2%

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

3. Applied egg-rr 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-udef 99.2%

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

   3. associate-+l+ 99.2%

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

   4. flip-+ 62.5%

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

   5. pow2 62.5%

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

   6. pow2 62.5%

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

   7. pow-sqr 62.5%

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

   8. metadata-eval 62.5%

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

5. Applied egg-rr 62.5%

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

   1. metadata-eval 62.5%

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

   2. pow-plus 62.5%

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

   3. unpow3 62.5%

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

   4. associate-*r* 62.5%

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

   5. flip-+ 99.2%

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

   6. +-commutative 99.2%

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

   7. metadata-eval 99.2%

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

   8. sub-neg 99.2%

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

   9. +-commutative 99.2%

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

   10. associate--r- 99.2%

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

   11. sub-neg 99.2%

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

   12. +-commutative 99.2%

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

   13. associate--r+ 99.2%

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

   14. distribute-rgt-neg-in 99.2%

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

7. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. cancel-sign-sub 99.2%

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

   3. +-commutative 99.2%

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

   4. distribute-lft1-in 99.2%

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

   5. +-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 97.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < -1 or 1 < N

   1. Initial program 50.8%

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

      1. diff-atan 60.6%

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

      2. +-commutative 60.6%

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

      3. associate--l+ 98.5%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

      6. distribute-lft1-in 98.5%

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

      7. associate-+r+ 98.5%

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

      8. +-commutative 98.5%

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

      9. +-commutative 98.5%

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

      10. fma-def 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in N around inf 95.4%

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

      1. unpow2 95.4%

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

   6. Simplified 95.4%

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

   if -1 < N < 1

   1. Initial program 100.0%

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

      1. diff-atan 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. associate-+r+ 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in N around 0 98.9%

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

4. Final simplification 97.0%

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

Alternative 3: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -0.62 \lor \neg \left(N \leq 1.6\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -0.62) (not (<= N 1.6)))
   (atan2 1.0 (* N N))
   (atan2 1.0 (+ N 1.0))))

C

double code(double N) {
	double tmp;
	if ((N <= -0.62) || !(N <= 1.6)) {
		tmp = atan2(1.0, (N * 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 <= (-0.62d0)) .or. (.not. (n <= 1.6d0))) then
        tmp = atan2(1.0d0, (n * 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 <= -0.62) || !(N <= 1.6)) {
		tmp = Math.atan2(1.0, (N * N));
	} else {
		tmp = Math.atan2(1.0, (N + 1.0));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -0.62) or not (N <= 1.6):
		tmp = math.atan2(1.0, (N * N))
	else:
		tmp = math.atan2(1.0, (N + 1.0))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -0.62) || !(N <= 1.6))
		tmp = atan(1.0, Float64(N * 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 <= -0.62) || ~((N <= 1.6)))
		tmp = atan2(1.0, (N * N));
	else
		tmp = atan2(1.0, (N + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -0.62], N[Not[LessEqual[N, 1.6]], $MachinePrecision]], N[ArcTan[1.0 / N[(N * N), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(N + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < -0.619999999999999996 or 1.6000000000000001 < N

   1. Initial program 50.8%

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

      1. diff-atan 60.6%

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

      2. +-commutative 60.6%

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

      3. associate--l+ 98.5%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

      6. distribute-lft1-in 98.5%

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

      7. associate-+r+ 98.5%

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

      8. +-commutative 98.5%

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

      9. +-commutative 98.5%

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

      10. fma-def 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in N around inf 95.4%

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

      1. unpow2 95.4%

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

   6. Simplified 95.4%

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

   if -0.619999999999999996 < N < 1.6000000000000001

   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. associate-+r+ 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in N around 0 99.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq -0.62 \lor \neg \left(N \leq 1.6\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + 1}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. diff-atan 78.8%

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

   2. +-commutative 78.8%

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

   3. associate--l+ 99.2%

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

   4. +-inverses 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. associate-+r+ 99.2%

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

   8. +-commutative 99.2%

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

   9. +-commutative 99.2%

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

   10. fma-def 99.2%

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

3. Applied egg-rr 99.2%

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

   1. remove-double-neg 99.2%

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

   2. sub-neg 99.2%

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

   3. fma-udef 99.2%

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

   4. +-commutative 99.2%

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

   5. distribute-neg-in 99.2%

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

   6. unsub-neg 99.2%

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

   7. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 97.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. diff-atan 78.8%

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

   2. +-commutative 78.8%

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

   3. associate--l+ 99.2%

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

   4. +-inverses 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. associate-+r+ 99.2%

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

   8. +-commutative 99.2%

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

   9. +-commutative 99.2%

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

   10. fma-def 99.2%

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

3. Applied egg-rr 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-udef 99.2%

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

   3. associate-+l+ 99.2%

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

   4. flip-+ 62.5%

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

   5. pow2 62.5%

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

   6. pow2 62.5%

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

   7. pow-sqr 62.5%

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

   8. metadata-eval 62.5%

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

5. Applied egg-rr 62.5%

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

   1. metadata-eval 62.5%

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

   2. pow-plus 62.5%

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

   3. unpow3 62.5%

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

   4. associate-*r* 62.5%

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

   5. flip-+ 99.2%

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

   6. +-commutative 99.2%

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

   7. metadata-eval 99.2%

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

   8. sub-neg 99.2%

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

   9. +-commutative 99.2%

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

   10. associate--r- 99.2%

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

   11. sub-neg 99.2%

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

   12. +-commutative 99.2%

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

   13. associate--r+ 99.2%

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

   14. distribute-rgt-neg-in 99.2%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in N around inf 97.0%

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

   1. unpow2 97.0%

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

10. Simplified 97.0%

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

11. Final simplification 97.0%

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

Alternative 6: 51.1% 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 73.5%

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

   1. diff-atan 78.8%

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

   2. +-commutative 78.8%

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

   3. associate--l+ 99.2%

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

   4. +-inverses 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. associate-+r+ 99.2%

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

   8. +-commutative 99.2%

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

   9. +-commutative 99.2%

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

   10. fma-def 99.2%

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

3. Applied egg-rr 99.2%

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

4. Taylor expanded in N around 0 48.0%

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

5. Final simplification 48.0%

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

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 2023297 
(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)))