Average Error: 15.0 → 0.3
Time: 3.2s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(1 + \left(\mathsf{fma}\left(N, N, N\right) + -1\right)\right)} \]
(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N)
 :precision binary64
 (atan2 1.0 (+ 1.0 (+ 1.0 (+ (fma N N N) -1.0)))))
double code(double N) {
	return atan((N + 1.0)) - atan(N);
}
↓
double code(double N) {
	return atan2(1.0, (1.0 + (1.0 + (fma(N, N, N) + -1.0))));
}

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

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

Error

Target

Original15.0
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

  1. Initial program 15.0

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

  2. Applied egg-rr14.0

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

  3. Taylor expanded in N around 0 0.3

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

  4. Applied egg-rr0.6

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{1 + \mathsf{fma}\left(N, N, N\right)}\right)}^{2}, \sqrt[3]{1 + \mathsf{fma}\left(N, N, N\right)}, -1\right)}} \]

  5. Applied egg-rr0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2022190 
(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)))