2atan (example 3.5)

Percentage Accurate: 77.3% → 99.4%

Time: 5.7s

Alternatives: 7

Speedup: 1.0×

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: 77.3% 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

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

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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (atan (+ 1.0 N)) (atan N))))
   (if (<= t_0 2e-10) (atan2 1.0 (fma N N N)) t_0)))

C

double code(double N) {
	double t_0 = atan((1.0 + N)) - atan(N);
	double tmp;
	if (t_0 <= 2e-10) {
		tmp = atan2(1.0, fma(N, N, 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 <= 2e-10)
		tmp = atan(1.0, fma(N, N, N));
	else
		tmp = t_0;
	end
	return 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, 2e-10], N[ArcTan[1.0 / N[(N * N + N), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq -0.62 \lor \neg \left(N \leq 1.62\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (or (<= N -0.62) (not (<= N 1.62)))
   (atan2 1.0 (pow N 2.0))
   (atan2 1.0 (+ 1.0 N))))

C

double code(double N) {
	double tmp;
	if ((N <= -0.62) || !(N <= 1.62)) {
		tmp = atan2(1.0, pow(N, 2.0));
	} else {
		tmp = atan2(1.0, (1.0 + N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.62d0)) .or. (.not. (n <= 1.62d0))) then
        tmp = atan2(1.0d0, (n ** 2.0d0))
    else
        tmp = atan2(1.0d0, (1.0d0 + n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((N <= -0.62) || !(N <= 1.62)) {
		tmp = Math.atan2(1.0, Math.pow(N, 2.0));
	} else {
		tmp = Math.atan2(1.0, (1.0 + N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (N <= -0.62) or not (N <= 1.62):
		tmp = math.atan2(1.0, math.pow(N, 2.0))
	else:
		tmp = math.atan2(1.0, (1.0 + N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if ((N <= -0.62) || !(N <= 1.62))
		tmp = atan(1.0, (N ^ 2.0));
	else
		tmp = atan(1.0, Float64(1.0 + N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((N <= -0.62) || ~((N <= 1.62)))
		tmp = atan2(1.0, (N ^ 2.0));
	else
		tmp = atan2(1.0, (1.0 + N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[Or[LessEqual[N, -0.62], N[Not[LessEqual[N, 1.62]], $MachinePrecision]], N[ArcTan[1.0 / N[Power[N, 2.0], $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0 / N[(1.0 + N), $MachinePrecision]], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 75.8% accurate, 1.9× speedup

Math

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

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(Float64(Float64(1.0 + N) - N), 1.0)
end

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 52.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 51.4% 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

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 2024003 
(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)))