2atan (example 3.5)

Average Accuracy: 77.4% → 99.4%

Time: 7.2s

Precision: binary64

Cost: 6912

Math

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

↓

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

FPCore

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

↓

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

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

↓

double code(double N) {
	return atan2(1.0, (1.0 + (N * (1.0 + N))));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan((n + 1.0d0)) - atan(n)
end function

↓

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan2(1.0d0, (1.0d0 + (n * (1.0d0 + n))))
end function

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

↓

public static double code(double N) {
	return Math.atan2(1.0, (1.0 + (N * (1.0 + N))));
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

↓

def code(N):
	return math.atan2(1.0, (1.0 + (N * (1.0 + 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 + Float64(N * Float64(1.0 + N))))
end

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

↓

function tmp = code(N)
	tmp = atan2(1.0, (1.0 + (N * (1.0 + 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[(1.0 + N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	77.4%
Target	99.4%
Herbie	99.4%

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

Derivation

1. Initial program 77.4%

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

2. Applied egg-rr 79.0%

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

   [Start] 77.4	\[ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
   diff-atan [=>] 79.1	\[ \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]
   associate--l+ [=>] 79.1	\[ \tan^{-1}_* \frac{\color{blue}{N + \left(1 - N\right)}}{1 + \left(N + 1\right) \cdot N} \]
   +-commutative [=>] 79.1	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]
   *-commutative [=>] 79.1	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{N \cdot \left(N + 1\right)} + 1} \]
   distribute-rgt-in [=>] 79.0	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\left(N \cdot N + 1 \cdot N\right)} + 1} \]
   *-un-lft-identity [<=] 79.0	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\left(N \cdot N + \color{blue}{N}\right) + 1} \]
   associate-+r+ [<=] 79.0	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{N \cdot N + \left(N + 1\right)}} \]
   +-commutative [=>] 79.0	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\color{blue}{\left(N + 1\right) + N \cdot N}} \]

3. Simplified 99.4%

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

   [Start] 79.0	\[ \tan^{-1}_* \frac{N + \left(1 - N\right)}{\left(N + 1\right) + N \cdot N} \]
   associate-+r- [=>] 79.1	\[ \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 79.1	\[ \tan^{-1}_* \frac{\color{blue}{\left(1 + N\right)} - N}{\left(N + 1\right) + N \cdot N} \]
   associate--l+ [=>] 99.4	\[ \tan^{-1}_* \frac{\color{blue}{1 + \left(N - N\right)}}{\left(N + 1\right) + N \cdot N} \]
   +-commutative [=>] 99.4	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\color{blue}{N \cdot N + \left(N + 1\right)}} \]
   fma-def [=>] 99.4	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\color{blue}{\mathsf{fma}\left(N, N, N + 1\right)}} \]
   +-commutative [=>] 99.4	\[ \tan^{-1}_* \frac{1 + \left(N - N\right)}{\mathsf{fma}\left(N, N, \color{blue}{1 + N}\right)} \]

4. Taylor expanded in N around 0 99.4%

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

5. Applied egg-rr 99.4%

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

   [Start] 99.4	\[ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N + 1\right)} \]
   fma-udef [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{N \cdot N + \left(N + 1\right)}} \]
   associate-+r+ [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{\left(N \cdot N + N\right) + 1}} \]

6. Applied egg-rr 99.4%

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

   [Start] 99.4	\[ \tan^{-1}_* \frac{1}{\left(N \cdot N + N\right) + 1} \]
   distribute-lft1-in [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{\left(N + 1\right) \cdot N} + 1} \]
   +-commutative [=>] 99.4	\[ \tan^{-1}_* \frac{1}{\color{blue}{\left(1 + N\right)} \cdot N + 1} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	97.4%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;N \leq -1 \lor \neg \left(N \leq 1\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1}\\ \end{array} \]

Alternative 2
Accuracy	98.0%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;N \leq -0.62 \lor \neg \left(N \leq 1.6\right):\\ \;\;\;\;\tan^{-1}_* \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{1}{1 + N}\\ \end{array} \]

Alternative 3
Accuracy	97.4%
Cost	6784

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

Alternative 4
Accuracy	52.5%
Cost	6528

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

Reproduce

herbie shell --seed 2023126 
(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)))