2atan (example 3.5)

Average Accuracy: 76.5% → 99.4%

Time: 5.6s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Target

Original	76.5%
Target	99.4%
Herbie	99.4%

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

Derivation

1. Initial program 76.5%

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

2. Applied egg-rr 78.3%

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

   [Start] 76.5	\[ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
   diff-atan [=>] 78.3	\[ \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]
   associate--l+ [=>] 78.3	\[ \tan^{-1}_* \frac{\color{blue}{N + \left(1 - N\right)}}{1 + \left(N + 1\right) \cdot N} \]
   +-commutative [=>] 78.3	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]
   *-commutative [=>] 78.3	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{N \cdot \left(N + 1\right)} + 1} \]
   fma-def [=>] 78.3	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\mathsf{fma}\left(N, N + 1, 1\right)}} \]

3. Simplified 99.4%

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

   [Start] 78.3	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   +-commutative [=>] 78.3	\[ \tan^{-1}_* \frac{\color{blue}{\left(1 - N\right) + N}}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   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. Taylor expanded in N around 0 99.4%

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

5. Simplified 99.4%

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

   [Start] 99.4	\[ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)} \]
   fma-udef [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{N \cdot \left(N + 1\right) + 1}} \]
   +-commutative [<=] 99.4	\[ \tan^{-1}_* \frac{1}{N \cdot \color{blue}{\left(1 + N\right)} + 1} \]
   distribute-rgt-in [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{\left(1 \cdot N + N \cdot N\right)} + 1} \]
   *-lft-identity [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\left(\color{blue}{N} + N \cdot N\right) + 1} \]
   associate-+l+ [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{N + \left(N \cdot N + 1\right)}} \]
   fma-def [=>] 99.4	\[ \tan^{-1}_* \frac{1}{N + \color{blue}{\mathsf{fma}\left(N, N, 1\right)}} \]

6. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	98.9%
Cost	26308

\[\begin{array}{l} t_0 := \tan^{-1} \left(1 + N\right) - \tan^{-1} N\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\tan^{-1}_* \frac{1}{N + N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	98.3%
Cost	13320

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

Alternative 3
Accuracy	98.3%
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 4
Accuracy	97.2%
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 5
Accuracy	97.7%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;N \leq -0.6 \lor \neg \left(N \leq 1.6\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \]

Alternative 6
Accuracy	52.0%
Cost	6528

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

Reproduce

herbie shell --seed 2023151 
(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)))