2atan (example 3.5)

Average Error: 15.2 → 0.4

Time: 8.5s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

(FPCore (N) :precision binary64 (atan2 1.0 (+ 1.0 (fma N N N))))

C

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)));
}

Julia

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

Wolfram

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]

Target

Original	15.2
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 15.2

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

2. Applied egg-rr 13.9

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

3. Taylor expanded in N around 0 0.4

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

4. Simplified 0.4

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}} \]
   Proof
   (atan2.f64 1 (+.f64 1 (fma.f64 N N N))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 1 (+.f64 1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 N N) N)))): 1 points increase in error, 0 points decrease in error
   (atan2.f64 1 (+.f64 1 (+.f64 (*.f64 N N) (Rewrite<= *-lft-identity_binary64 (*.f64 1 N))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 1 (+.f64 1 (Rewrite<= distribute-rgt-in_binary64 (*.f64 N (+.f64 N 1))))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.6
Cost	6920

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{if}\;N \leq -141.95970556005847:\\ \;\;\;\;t_0\\ \mathbf{elif}\;N \leq 0.007115602309660742:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.4
Cost	6912

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

Alternative 3
Error	1.9
Cost	6784

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

Alternative 4
Error	30.4
Cost	6656

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

Alternative 5
Error	31.2
Cost	6528

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

Reproduce

herbie shell --seed 2022316 
(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)))