2atan (example 3.5)

Average Error: 15.0 → 0.4

Time: 6.5s

Precision: binary64

Cost: 6912

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double N) {
	return atan2(1.0, (1.0 + (N * (1.0 + 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 * (1.0d0 + 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 * (1.0 + 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 * (1.0 + N))))

Julia

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

↓

function code(N)
	return atan(1.0, Float64(1.0 + Float64(N * Float64(1.0 + 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 * (1.0 + 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[(1.0 + N[(N * N[(1.0 + N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	15.0
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 15.0

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

2. Applied egg-rr 23.6

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1} \left(N + 1\right)\right)\right)} - \tan^{-1} N \]

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

   [Start] 0.4	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 0.4	\[ \tan^{-1}_* \frac{\color{blue}{\left(N - N\right) + 1}}{\left(N + 1\right) + N \cdot N} \]
   +-inverses [=>] 0.4	\[ \tan^{-1}_* \frac{\color{blue}{0} + 1}{\left(N + 1\right) + N \cdot N} \]
   metadata-eval [=>] 0.4	\[ \tan^{-1}_* \frac{\color{blue}{1}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 0.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{N \cdot N + \left(N + 1\right)}} \]
   fma-def [=>] 0.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(N, N, N + 1\right)}} \]

5. Applied egg-rr 0.4

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.6
Cost	6921

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

Alternative 2
Error	1.3
Cost	6921

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

Alternative 3
Error	31.1
Cost	6528

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

Reproduce

herbie shell --seed 2023018 
(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)))