Average Error: 63.0 → 0
Time: 4.3s
Precision: binary64
\[n > 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]
↓
\[\log n + \frac{0.5}{n} \]
(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))
↓
(FPCore (n) :precision binary64 (+ (log n) (/ 0.5 n)))
double code(double n) {
	return (((n + 1.0) * log((n + 1.0))) - (n * log(n))) - 1.0;
}
↓
double code(double n) {
	return log(n) + (0.5 / n);
}

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((n + 1.0d0) * log((n + 1.0d0))) - (n * log(n))) - 1.0d0
end function
↓
real(8) function code(n)
    real(8), intent (in) :: n
    code = log(n) + (0.5d0 / n)
end function

public static double code(double n) {
	return (((n + 1.0) * Math.log((n + 1.0))) - (n * Math.log(n))) - 1.0;
}
↓
public static double code(double n) {
	return Math.log(n) + (0.5 / n);
}

def code(n):
	return (((n + 1.0) * math.log((n + 1.0))) - (n * math.log(n))) - 1.0
↓
def code(n):
	return math.log(n) + (0.5 / n)

function code(n)
	return Float64(Float64(Float64(Float64(n + 1.0) * log(Float64(n + 1.0))) - Float64(n * log(n))) - 1.0)
end
↓
function code(n)
	return Float64(log(n) + Float64(0.5 / n))
end

function tmp = code(n)
	tmp = (((n + 1.0) * log((n + 1.0))) - (n * log(n))) - 1.0;
end
↓
function tmp = code(n)
	tmp = log(n) + (0.5 / n);
end

code[n_] := N[(N[(N[(N[(n + 1.0), $MachinePrecision] * N[Log[N[(n + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(n * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[n_] := N[(N[Log[n], $MachinePrecision] + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]

\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log n + \frac{0.5}{n}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right) \]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]

  2. Simplified44.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(n\right) - \mathsf{fma}\left(n, \log n - \mathsf{log1p}\left(n\right), 1\right)} \]

  3. Taylor expanded in n around inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} + -1 \cdot \log \left(\frac{1}{n}\right)} \]

  4. Simplified0

    \[\leadsto \color{blue}{\log n + \frac{0.5}{n}} \]

  5. Final simplification0

    \[\leadsto \log n + \frac{0.5}{n} \]

Reproduce

herbie shell --seed 2022212 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))