Average Error: 15.3 → 0.4
Time: 3.1s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1 + N\right)} \]
(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N) :precision binary64 (atan2 1.0 (fma N N (+ 1.0 N))))
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)));
}

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

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]

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

Error

Target

Original15.3
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

  1. Initial program 15.3

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

  2. Applied egg-rr14.2

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

  3. Taylor expanded in N around 0 0.4

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

  4. Taylor expanded in N around inf 0.4

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{1}{N + \left(1 + {N}^{2}\right)}} \]

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2022192 
(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)))