2atan (example 3.5)

Average Error: 15.4 → 15.4

Time: 7.9s

Precision: binary64

Cost: 26432

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double N) {
	return ((atan((N + 1.0)) + (atan(N) * 0.5)) - atan(N)) + (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
    code = ((atan((n + 1.0d0)) + (atan(n) * 0.5d0)) - atan(n)) + (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) {
	return ((Math.atan((N + 1.0)) + (Math.atan(N) * 0.5)) - Math.atan(N)) + (Math.atan(N) * -0.5);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[N_] := N[(N[(N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[ArcTan[N], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[N], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Target

Original	15.4
Target	0.4
Herbie	15.4

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

Derivation

1. Initial program 15.4

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

2. Applied egg-rr 15.4

   \[\leadsto \color{blue}{\left(\left(\tan^{-1} \left(N + 1\right) + \tan^{-1} N\right) - \left(0 - \tan^{-1} N \cdot -1.5\right)\right) + \tan^{-1} N \cdot -0.5} \]

3. Applied egg-rr 15.5

   \[\leadsto \color{blue}{\left(\tan^{-1} N \cdot -2.5 + \left(\tan^{-1} \left(N + 1\right) + \tan^{-1} N \cdot 2\right)\right)} + \tan^{-1} N \cdot -0.5 \]

4. Applied egg-rr 15.4

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

5. Final simplification 15.4

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

Alternatives

Alternative 1
Error	15.4
Cost	19904

\[\tan^{-1} \left(N + 1\right) - \left(\tan^{-1} N \cdot 1.75 - \tan^{-1} N \cdot 0.75\right) \]

Alternative 2
Error	15.4
Cost	19904

\[\left(\tan^{-1} \left(N + 1\right) + \tan^{-1} N \cdot -0.75\right) - \tan^{-1} N \cdot 0.25 \]

Alternative 3
Error	15.4
Cost	13120

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

Reproduce

herbie shell --seed 2023101 
(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)))