2atan (example 3.5)

Average Error: 22.73% → 0.45%

Time: 8.3s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double N) {
	return atan2(1.0, fma(N, 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, fma(N, N, Float64(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[(N * N + N[(1.0 + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	22.73%
Target	0.45%
Herbie	0.45%

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

Derivation

1. Initial program 22.73

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

2. Applied egg-rr 20.92

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

3. Simplified 0.45

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

   [Start] 20.92	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\left(N + 1\right) + N \cdot N} \]
   associate-+r- [=>] 20.91	\[ \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 20.91	\[ \tan^{-1}_* \frac{\color{blue}{\left(1 + N\right)} - N}{\left(N + 1\right) + N \cdot N} \]
   associate--l+ [=>] 0.45	\[ \tan^{-1}_* \frac{\color{blue}{1 + \left(N - N\right)}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 0.45	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\color{blue}{N \cdot N + \left(N + 1\right)}} \]
   fma-def [=>] 0.45	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\color{blue}{\mathsf{fma}\left(N, N, N + 1\right)}} \]
   +-commutative [=>] 0.45	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)} \]

4. Taylor expanded in N around 0 0.45

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

5. Final simplification 0.45

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

Alternatives

Alternative 1
Error	1.4%
Cost	7049

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

Alternative 2
Error	2.47%
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 3
Error	1.92%
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 4
Error	0.45%
Cost	6912

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

Alternative 5
Error	48.46%
Cost	6528

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

Reproduce

herbie shell --seed 2023125 
(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)))