2atan (example 3.5)

Average Error: 15.0 → 15.0

Time: 19.5s

Precision: binary64

Cost: 26688

Math

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

↓

\[\begin{array}{l} t_0 := \tan^{-1} \left(N + 1\right)\\ \left(\left(\left(\left(1 + t_0\right) + -1\right) - \left(\tan^{-1} N - t_0\right)\right) - \tan^{-1} N\right) \cdot 0.5 \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (let* ((t_0 (atan (+ N 1.0))))
   (* (- (- (+ (+ 1.0 t_0) -1.0) (- (atan N) t_0)) (atan N)) 0.5)))

C

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

↓

double code(double N) {
	double t_0 = atan((N + 1.0));
	return ((((1.0 + t_0) + -1.0) - (atan(N) - t_0)) - atan(N)) * 0.5;
}

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
    real(8) :: t_0
    t_0 = atan((n + 1.0d0))
    code = ((((1.0d0 + t_0) + (-1.0d0)) - (atan(n) - t_0)) - atan(n)) * 0.5d0
end function

Java

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

↓

public static double code(double N) {
	double t_0 = Math.atan((N + 1.0));
	return ((((1.0 + t_0) + -1.0) - (Math.atan(N) - t_0)) - Math.atan(N)) * 0.5;
}

Python

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

↓

def code(N):
	t_0 = math.atan((N + 1.0))
	return ((((1.0 + t_0) + -1.0) - (math.atan(N) - t_0)) - math.atan(N)) * 0.5

Julia

function code(N)
	return Float64(atan(Float64(N + 1.0)) - atan(N))
end

↓

function code(N)
	t_0 = atan(Float64(N + 1.0))
	return Float64(Float64(Float64(Float64(Float64(1.0 + t_0) + -1.0) - Float64(atan(N) - t_0)) - atan(N)) * 0.5)
end

MATLAB

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

↓

function tmp = code(N)
	t_0 = atan((N + 1.0));
	tmp = ((((1.0 + t_0) + -1.0) - (atan(N) - t_0)) - atan(N)) * 0.5;
end

Wolfram

code[N_] := N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]

↓

code[N_] := Block[{t$95$0 = N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] - N[(N[ArcTan[N], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Target

Original	15.0
Target	0.5
Herbie	15.0

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

Derivation

1. Initial program 15.0

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

2. Applied egg-rr 15.0

   \[\leadsto \color{blue}{\left(\left(\tan^{-1} \left(N + 1\right) - \tan^{-1} N\right) \cdot 0.5 + -1\right) + \left(1 + \left(\tan^{-1} \left(N + 1\right) - \tan^{-1} N\right) \cdot 0.5\right)} \]

3. Applied egg-rr 15.0

   \[\leadsto \left(\left(\tan^{-1} \left(N + 1\right) - \tan^{-1} N\right) \cdot 0.5 + -1\right) + \color{blue}{\frac{\tan^{-1} \left(N + 1\right) + \left(2 - \tan^{-1} N\right)}{2}} \]

4. Applied egg-rr 15.0

   \[\leadsto \color{blue}{\left(\left(\tan^{-1} \left(N + 1\right) + \left(2 - \tan^{-1} N\right)\right) + \left(\left(\tan^{-1} \left(N + 1\right) - \tan^{-1} N\right) + -2\right)\right) \cdot 0.5} \]

5. Applied egg-rr 15.0

   \[\leadsto \color{blue}{\left(\left(\left(\left(1 + \tan^{-1} \left(N + 1\right)\right) + -1\right) - \left(\tan^{-1} N - \tan^{-1} \left(N + 1\right)\right)\right) - \tan^{-1} N\right)} \cdot 0.5 \]

6. Final simplification 15.0

   \[\leadsto \left(\left(\left(\left(1 + \tan^{-1} \left(N + 1\right)\right) + -1\right) - \left(\tan^{-1} N - \tan^{-1} \left(N + 1\right)\right)\right) - \tan^{-1} N\right) \cdot 0.5 \]

Alternatives

Alternative 1
Error	15.0
Cost	13376

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

Alternative 2
Error	15.0
Cost	13120

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

Reproduce

herbie shell --seed 2023073 
(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)))