logs (example 3.8)

Percentage Accurate: 1.6% → 100.0%

Time: 8.5s

Precision: binary64

Cost: 6464

\[n > 6.8 \cdot 10^{+15}\]

Math

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

↓

\[\log n \]

FPCore

(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))

↓

(FPCore (n) :precision binary64 (log n))

C

double code(double n) {
	return (((n + 1.0) * log((n + 1.0))) - (n * log(n))) - 1.0;
}

↓

double code(double n) {
	return log(n);
}

Fortran

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

Java

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);
}

Python

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)

Julia

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

MATLAB

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

Wolfram

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]

Target

Original	1.6%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 1.5%

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

2. Simplified 3.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), -1\right) - n \cdot \log n} \]
   Step-by-step derivation

   [Start] 1.5	\[ \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]
   sub-neg [=>] 1.5	\[ \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) + \left(-1\right)} \]
   +-commutative [=>] 1.5	\[ \color{blue}{\left(-1\right) + \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right)} \]
   associate-+r- [=>] 3.0	\[ \color{blue}{\left(\left(-1\right) + \left(n + 1\right) \cdot \log \left(n + 1\right)\right) - n \cdot \log n} \]
   +-commutative [<=] 3.0	\[ \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) + \left(-1\right)\right)} - n \cdot \log n \]
   fma-def [=>] 3.0	\[ \color{blue}{\mathsf{fma}\left(n + 1, \log \left(n + 1\right), -1\right)} - n \cdot \log n \]
   +-commutative [=>] 3.0	\[ \mathsf{fma}\left(n + 1, \log \color{blue}{\left(1 + n\right)}, -1\right) - n \cdot \log n \]
   log1p-def [=>] 3.0	\[ \mathsf{fma}\left(n + 1, \color{blue}{\mathsf{log1p}\left(n\right)}, -1\right) - n \cdot \log n \]
   metadata-eval [=>] 3.0	\[ \mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), \color{blue}{-1}\right) - n \cdot \log n \]

3. Taylor expanded in n around inf 100.0%

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

4. Simplified 100.0%

   \[\leadsto \color{blue}{\log n} \]
   Step-by-step derivation

   [Start] 100.0	\[ -1 \cdot \log \left(\frac{1}{n}\right) \]
   mul-1-neg [=>] 100.0	\[ \color{blue}{-\log \left(\frac{1}{n}\right)} \]
   log-rec [=>] 100.0	\[ -\color{blue}{\left(-\log n\right)} \]
   remove-double-neg [=>] 100.0	\[ \color{blue}{\log n} \]

5. Final simplification 100.0%

   \[\leadsto \log n \]

Alternatives

Alternative 1
Accuracy	3.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023161 
(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))