2atan (example 3.5)

Average Error: 15.2 → 15.2

Time: 14.0s

Precision: binary64

Cost: 20544

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = ((-1.0d0) - atan(n)) * 0.5d0
    code = (atan((n + 1.0d0)) + ((t_0 + (-1.0d0)) + (1.0d0 + t_0))) + 1.0d0
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 = (-1.0 - Math.atan(N)) * 0.5;
	return (Math.atan((N + 1.0)) + ((t_0 + -1.0) + (1.0 + t_0))) + 1.0;
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(N)
	t_0 = (-1.0 - atan(N)) * 0.5;
	tmp = (atan((N + 1.0)) + ((t_0 + -1.0) + (1.0 + t_0))) + 1.0;
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[(N[(-1.0 - N[ArcTan[N], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$0 + -1.0), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Target

Original	15.2
Target	0.4
Herbie	15.2

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

Derivation

1. Initial program 15.2

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

2. Applied egg-rr 15.2

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

3. Applied egg-rr 15.2

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

4. Final simplification 15.2

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

Alternatives

Alternative 1
Error	15.2
Cost	13376

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

Alternative 2
Error	15.2
Cost	13120

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

Reproduce

herbie shell --seed 2023074 
(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)))