logs (example 3.8)

Average Error: 63.0 → 0

Time: 7.1s

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	63.0
Target	0.0
Herbie	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 63.0

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

2. Simplified 62.0

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

   [Start] 63.0	\[ \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]
   metadata-eval [<=] 63.0	\[ \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - \color{blue}{\left(1 + 0\right)} \]
   associate--l- [<=] 63.0	\[ \color{blue}{\left(\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\right) - 0} \]
   associate--l- [=>] 62.0	\[ \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - \left(n \cdot \log n + 1\right)\right)} - 0 \]
   sub-neg [=>] 62.0	\[ \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) + \left(-\left(n \cdot \log n + 1\right)\right)\right)} - 0 \]
   associate--l+ [=>] 62.0	\[ \color{blue}{\left(n + 1\right) \cdot \log \left(n + 1\right) + \left(\left(-\left(n \cdot \log n + 1\right)\right) - 0\right)} \]
   +-commutative [<=] 62.0	\[ \color{blue}{\left(\left(-\left(n \cdot \log n + 1\right)\right) - 0\right) + \left(n + 1\right) \cdot \log \left(n + 1\right)} \]
   associate--r- [<=] 62.0	\[ \color{blue}{\left(-\left(n \cdot \log n + 1\right)\right) - \left(0 - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)} \]
   neg-sub0 [<=] 62.0	\[ \left(-\left(n \cdot \log n + 1\right)\right) - \color{blue}{\left(-\left(n + 1\right) \cdot \log \left(n + 1\right)\right)} \]
   sub-neg [=>] 62.0	\[ \color{blue}{\left(-\left(n \cdot \log n + 1\right)\right) + \left(-\left(-\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)} \]
   distribute-neg-in [<=] 62.0	\[ \color{blue}{-\left(\left(n \cdot \log n + 1\right) + \left(-\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)} \]
   +-commutative [=>] 62.0	\[ -\color{blue}{\left(\left(-\left(n + 1\right) \cdot \log \left(n + 1\right)\right) + \left(n \cdot \log n + 1\right)\right)} \]
   distribute-neg-in [=>] 62.0	\[ \color{blue}{\left(-\left(-\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right) + \left(-\left(n \cdot \log n + 1\right)\right)} \]

3. Taylor expanded in n around inf 0.0

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

4. Simplified 0

   \[\leadsto \color{blue}{\log n} \]
   Proof

   [Start] 0.0	\[ -1 \cdot \log \left(\frac{1}{n}\right) \]
   mul-1-neg [=>] 0.0	\[ \color{blue}{-\log \left(\frac{1}{n}\right)} \]
   log-rec [=>] 0	\[ -\color{blue}{\left(-\log n\right)} \]
   remove-double-neg [=>] 0	\[ \color{blue}{\log n} \]

5. Final simplification 0

   \[\leadsto \log n \]

Alternatives

Alternative 1
Error	63.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023011 
(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))