2atan (example 3.5)

Average Accuracy: 76.8% → 99.4%

Time: 4.5s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Target

Original	76.8%
Target	99.4%
Herbie	99.4%

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

Derivation

1. Initial program 76.8%

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

2. Applied egg-rr 78.5%

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

   [Start] 76.8	\[ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
   diff-atan [=>] 78.5	\[ \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]
   associate--l+ [=>] 78.5	\[ \tan^{-1}_* \frac{\color{blue}{N + \left(1 - N\right)}}{1 + \left(N + 1\right) \cdot N} \]
   +-commutative [=>] 78.5	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]
   *-commutative [=>] 78.5	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{N \cdot \left(N + 1\right)} + 1} \]
   fma-def [=>] 78.5	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\mathsf{fma}\left(N, N + 1, 1\right)}} \]

3. Simplified 99.4%

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

   [Start] 78.5	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   +-commutative [=>] 78.5	\[ \tan^{-1}_* \frac{\color{blue}{\left(1 - N\right) + N}}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   associate-+l- [=>] 99.4	\[ \tan^{-1}_* \frac{\color{blue}{1 - \left(N - N\right)}}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   +-inverses [=>] 99.4	\[ \tan^{-1}_* \frac{1 - \color{blue}{0}}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   metadata-eval [=>] 99.4	\[ \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   +-commutative [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \color{blue}{1 + N}, 1\right)} \]

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	76.8%
Cost	13120

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

Reproduce

herbie shell --seed 2023152 
(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)))