Average Error: 14.7 → 0.3
Time: 2.4s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}\right)\right) \]
(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N)
 :precision binary64
 (log1p (expm1 (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 log1p(expm1(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 log1p(expm1(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[Log[1 + N[(Exp[N[ArcTan[1.0 / N[(1.0 + N[(N * N + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}\right)\right)

Error

Bits error versus N

Target

Original14.7
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

  1. Initial program 14.7

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

  2. Applied egg-rr0.3

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

  3. Applied egg-rr0.6

    \[\leadsto \tan^{-1}_* \frac{1 + \left(N - N\right)}{1 + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(N, N, N\right)}\right)}^{3}}} \]

  4. Applied egg-rr0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N\right) + 1}\right)\right)} \]

  5. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022150 
(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)))