logs (example 3.8)

Average Error: 63.0 → 0.0

Time: 7.5s

Precision: binary64

Cost: 13376

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

Math

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

↓

\[\mathsf{log1p}\left(n\right) + \left(n \cdot \mathsf{log1p}\left(\frac{1}{n}\right) + -1\right) \]

FPCore

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

↓

(FPCore (n) :precision binary64 (+ (log1p n) (+ (* n (log1p (/ 1.0 n))) -1.0)))

C

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

↓

double code(double n) {
	return log1p(n) + ((n * log1p((1.0 / n))) + -1.0);
}

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.log1p(n) + ((n * Math.log1p((1.0 / n))) + -1.0);
}

Python

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

↓

def code(n):
	return math.log1p(n) + ((n * math.log1p((1.0 / n))) + -1.0)

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 Float64(log1p(n) + Float64(Float64(n * log1p(Float64(1.0 / n))) + -1.0))
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[(N[Log[1 + n], $MachinePrecision] + N[(N[(n * N[Log[1 + N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Target

Original	63.0
Target	0
Herbie	0.0

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

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 44.2

   \[\leadsto \color{blue}{\mathsf{log1p}\left(n\right) + \mathsf{fma}\left(n, \mathsf{log1p}\left(n\right) - \log n, -1\right)} \]
   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): 0 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. Applied egg-rr 44.2

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

4. Simplified 0.0

   \[\leadsto \mathsf{log1p}\left(n\right) + \mathsf{fma}\left(n, \color{blue}{\mathsf{log1p}\left({n}^{-1}\right)}, -1\right) \]
   Proof
   (log1p.f64 (pow.f64 n -1)): 0 points increase in error, 0 points decrease in error
   (log1p.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 n) -1)))): 131 points increase in error, 125 points decrease in error
   (log1p.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (log.f64 n))))): 0 points increase in error, 0 points decrease in error
   (log1p.f64 (exp.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (log.f64 n))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 (neg.f64 (log.f64 n)))))): 256 points increase in error, 0 points decrease in error
   (log.f64 (+.f64 (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 n) n)) (exp.f64 (neg.f64 (log.f64 n))))): 38 points increase in error, 0 points decrease in error
   (log.f64 (+.f64 (*.f64 (/.f64 1 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 n)))) n) (exp.f64 (neg.f64 (log.f64 n))))): 131 points increase in error, 125 points decrease in error
   (log.f64 (+.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (log.f64 n)))) n) (exp.f64 (neg.f64 (log.f64 n))))): 22 points increase in error, 24 points decrease in error
   (log.f64 (+.f64 (*.f64 (exp.f64 (neg.f64 (log.f64 n))) n) (Rewrite<= *-rgt-identity_binary64 (*.f64 (exp.f64 (neg.f64 (log.f64 n))) 1)))): 0 points increase in error, 0 points decrease in error
   (log.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 (exp.f64 (neg.f64 (log.f64 n))) (+.f64 n 1)))): 0 points increase in error, 1 points decrease in error
   (log.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 n 1) (exp.f64 (neg.f64 (log.f64 n)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> log-prod_binary64 (+.f64 (log.f64 (+.f64 n 1)) (log.f64 (exp.f64 (neg.f64 (log.f64 n)))))): 125 points increase in error, 131 points decrease in error
   (+.f64 (log.f64 (Rewrite=> +-commutative_binary64 (+.f64 1 n))) (log.f64 (exp.f64 (neg.f64 (log.f64 n))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> log1p-def_binary64 (log1p.f64 n)) (log.f64 (exp.f64 (neg.f64 (log.f64 n))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log1p.f64 n) (Rewrite=> rem-log-exp_binary64 (neg.f64 (log.f64 n)))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.0

   \[\leadsto \mathsf{log1p}\left(n\right) + \color{blue}{\left(n \cdot \mathsf{log1p}\left(\frac{1}{n}\right) + -1\right)} \]

6. Final simplification 0.0

   \[\leadsto \mathsf{log1p}\left(n\right) + \left(n \cdot \mathsf{log1p}\left(\frac{1}{n}\right) + -1\right) \]

Reproduce

herbie shell --seed 2022334 
(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))