2atan (example 3.5)

Average Error: 15.1 → 0.3

Time: 6.8s

Precision: binary64

Cost: 6912

Math

\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	15.1
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 15.1

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]

2. Applied egg-rr 14.0

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

3. Taylor expanded in N around 0 0.3

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

4. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022343 
(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)))