Average Error: 14.3 → 0.4
Time: 2.5s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
↓
\[\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)} \]
(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N) :precision binary64 (atan2 1.0 (+ 1.0 (fma N N N))))
double code(double N) {
	return atan((N + 1.0)) - atan(N);
}
↓
double code(double N) {
	return atan2(1.0, (1.0 + fma(N, N, N)));
}

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

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

Error

Bits error versus N

Target

Original14.3
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

  1. Initial program 14.3

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

  2. Applied egg-rr13.2

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

  3. Taylor expanded in N around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022166 
(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)))