2atan (example 3.5)

Average Error: 77.6% → 99.4%

Time: 8.5s

Precision: binary64

Cost: 13952.00

Math

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

↓

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

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (atan2 (+ 1.0 (- N N)) (+ 1.0 (* (fma N N -1.0) (* N (/ -1.0 (- 1.0 N)))))))

C

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

↓

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

Julia

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

↓

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

Target

Original	77.6%
Target	99.5%
Herbie	99.4%

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

Derivation

1. Initial program 77.6

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

2. Applied egg-rr 79.3

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

3. Simplified 99.5

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

   [Start] 79.3	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\left(N + 1\right) + N \cdot N} \]
   associate-+r- [=>] 79.3	\[ \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 79.3	\[ \tan^{-1}_* \frac{\color{blue}{\left(1 + N\right)} - N}{\left(N + 1\right) + N \cdot N} \]
   associate--l+ [=>] 99.5	\[ \tan^{-1}_* \frac{\color{blue}{1 + \left(N - N\right)}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 99.5	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\color{blue}{N \cdot N + \left(N + 1\right)}} \]
   fma-def [=>] 99.5	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\color{blue}{\mathsf{fma}\left(N, N, N + 1\right)}} \]
   +-commutative [=>] 99.5	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)} \]

4. Applied egg-rr 99.5

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

5. Applied egg-rr 92.3

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

6. Applied egg-rr 99.4

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

7. Final simplification 99.4

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

Alternatives

Alternative 1
Error	99.5%
Cost	6912.00

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

Alternative 2
Error	76.1%
Cost	6784.00

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

Alternative 3
Error	51.7%
Cost	6528.00

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

Reproduce

herbie shell --seed 2023097 
(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)))