Average Error: 63.0 → 0
Time: 5.9s
Precision: binary64
Cost: 6464
\[n > 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]
↓
\[\log n \]
(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))
↓
(FPCore (n) :precision binary64 (log 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);
}

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)
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);
}

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)

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 log(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);
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[Log[n], $MachinePrecision]

\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log n

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
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

    \[\leadsto \color{blue}{\mathsf{log1p}\left(n\right) + \mathsf{fma}\left(n, \mathsf{log1p}\left(n\right) - \log n, -1\right)} \]
    Proof
    (+.f64 (log1p.f64 n) (fma.f64 n (-.f64 (log1p.f64 n) (log.f64 n)) -1)): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 n))) (fma.f64 n (-.f64 (log1p.f64 n) (log.f64 n)) -1)): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 n 1))) (fma.f64 n (-.f64 (log1p.f64 n) (log.f64 n)) -1)): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (fma.f64 n (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 n))) (log.f64 n)) -1)): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (fma.f64 n (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 n 1))) (log.f64 n)) -1)): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (fma.f64 n (-.f64 (log.f64 (+.f64 n 1)) (log.f64 n)) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 n (-.f64 (log.f64 (+.f64 n 1)) (log.f64 n))) 1))): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (-.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (*.f64 (log.f64 n) n))) 1)): 2 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (-.f64 (-.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (Rewrite<= *-commutative_binary64 (*.f64 n (log.f64 n)))) 1)): 0 points increase in error, 0 points decrease in error
    (+.f64 (log.f64 (+.f64 n 1)) (-.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (*.f64 (neg.f64 n) (log.f64 n)))) 1)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (log.f64 (+.f64 n 1)) (+.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (*.f64 (neg.f64 n) (log.f64 n)))) 1)): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (*.f64 (neg.f64 n) (log.f64 n))) (log.f64 (+.f64 n 1)))) 1): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate--l+_binary64 (+.f64 (+.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (*.f64 (neg.f64 n) (log.f64 n))) (-.f64 (log.f64 (+.f64 n 1)) 1))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (neg.f64 n) (log.f64 n)) (*.f64 (log.f64 (+.f64 n 1)) n))) (-.f64 (log.f64 (+.f64 n 1)) 1)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 (*.f64 (neg.f64 n) (log.f64 n)) (*.f64 (log.f64 (+.f64 n 1)) n)) (log.f64 (+.f64 n 1))) 1)): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (neg.f64 n) (log.f64 n)) (+.f64 (*.f64 (log.f64 (+.f64 n 1)) n) (log.f64 (+.f64 n 1))))) 1): 254 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (*.f64 (neg.f64 n) (log.f64 n)) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 n (log.f64 (+.f64 n 1)))) (log.f64 (+.f64 n 1)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (*.f64 (neg.f64 n) (log.f64 n)) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 n 1) (log.f64 (+.f64 n 1))))) 1): 1 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (+.f64 n 1) (log.f64 (+.f64 n 1))) (*.f64 (neg.f64 n) (log.f64 n)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 (+.f64 n 1) (log.f64 (+.f64 n 1))) (*.f64 n (log.f64 n)))) 1): 0 points increase in error, 0 points decrease in error

  3. Taylor expanded in n around inf 0.0

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

  4. Simplified0

    \[\leadsto \color{blue}{\log n} \]
    Proof
    (log.f64 n): 0 points increase in error, 0 points decrease in error
    (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 n)))): 0 points increase in error, 0 points decrease in error
    (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 n)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 n)))): 0 points increase in error, 0 points decrease in error

  5. Final simplification0

    \[\leadsto \log n \]

Reproduce

herbie shell --seed 2022312 
(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))