Average Error: 29.3 → 0.1
Time: 6.6s
Precision: binary64
Cost: 19908
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 4e-6)
   (* (/ 1.0 N) (+ 1.0 (/ (+ (/ 0.3333333333333333 N) -0.5) N)))
   (log (/ (+ N 1.0) N))))
double code(double N) {
	return log((N + 1.0)) - log(N);
}
↓
double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 4e-6) {
		tmp = (1.0 / N) * (1.0 + (((0.3333333333333333 / N) + -0.5) / N));
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

real(8) function code(n)
    real(8), intent (in) :: n
    code = log((n + 1.0d0)) - log(n)
end function
↓
real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 4d-6) then
        tmp = (1.0d0 / n) * (1.0d0 + (((0.3333333333333333d0 / n) + (-0.5d0)) / n))
    else
        tmp = log(((n + 1.0d0) / n))
    end if
    code = tmp
end function

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}
↓
public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 4e-6) {
		tmp = (1.0 / N) * (1.0 + (((0.3333333333333333 / N) + -0.5) / N));
	} else {
		tmp = Math.log(((N + 1.0) / N));
	}
	return tmp;
}

def code(N):
	return math.log((N + 1.0)) - math.log(N)
↓
def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 4e-6:
		tmp = (1.0 / N) * (1.0 + (((0.3333333333333333 / N) + -0.5) / N))
	else:
		tmp = math.log(((N + 1.0) / N))
	return tmp

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end
↓
function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 4e-6)
		tmp = Float64(Float64(1.0 / N) * Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / N) + -0.5) / N)));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end
↓
function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 4e-6)
		tmp = (1.0 / N) * (1.0 + (((0.3333333333333333 / N) + -0.5) / N));
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]
↓
code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(1.0 / N), $MachinePrecision] * N[(1.0 + N[(N[(N[(0.3333333333333333 / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 4 \cdot 10^{-6}:\\
\;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 3.99999999999999982e-6

    1. Initial program 59.6

      \[\log \left(N + 1\right) - \log N \]

    2. Simplified59.6

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      Proof
      (-.f64 (log1p.f64 N) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 N))) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 N 1))) (log.f64 N)): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{N} + \frac{1}{N \cdot N} \cdot \left(\frac{0.3333333333333333}{N} - 0.5\right)} \]
      Proof
      (+.f64 (/.f64 1 N) (*.f64 (/.f64 1 (*.f64 N N)) (-.f64 (/.f64 1/3 N) 1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (*.f64 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 N 2))) (-.f64 (/.f64 1/3 N) 1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (/.f64 1/3 N) (/.f64 1 (pow.f64 N 2))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 2))) N)) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 6 points increase in error, 11 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/3 (/.f64 (/.f64 1 (pow.f64 N 2)) N))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 8 points increase in error, 8 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 (pow.f64 N 2) N)))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 7 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (/.f64 1 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 N N)) N))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (/.f64 1 (Rewrite<= unpow3_binary64 (pow.f64 N 3)))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 5 points increase in error, 4 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 1 N) (*.f64 1/3 (/.f64 1 (pow.f64 N 3)))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 0 points increase in error, 1 points decrease in error

    5. Applied egg-rr0.0

      \[\leadsto \frac{1}{N} + \color{blue}{\frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}} \]

    6. Applied egg-rr0.0

      \[\leadsto \color{blue}{\frac{1}{N} \cdot \left(\frac{\frac{0.3333333333333333}{N} + -0.5}{N} + 1\right)} \]

    if 3.99999999999999982e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.2

      \[\log \left(N + 1\right) - \log N \]

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      Proof
      (-.f64 (log1p.f64 N) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 N))) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 N 1))) (log.f64 N)): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error31.0
Cost192
\[\frac{1}{N} \]
Alternative 2
Error61.3
Cost64
\[0 \]
Alternative 3
Error61.1
Cost64
\[N \]

Error

Reproduce

herbie shell --seed 2022289 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))