2atan (example 3.5)

Average Error: 14.7 → 14.7

Time: 17.4s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	14.7
Target	0.4
Herbie	14.7

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

Derivation

1. Initial program 14.7

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

2. Applied egg-rr 14.7

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

3. Final simplification 14.7

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

Alternatives

Alternative 1
Error	14.7
Cost	13120

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

Reproduce

herbie shell --seed 2023075 
(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)))