?

Average Error: 3.1 → 0.2
Time: 7.7s
Precision: binary64
Cost: 6592

?

\[0.01 \leq x \land x \leq 1000\]
\[\log \left(x + 1\right) - \log x \]
↓
\[\mathsf{log1p}\left(\frac{1}{x}\right) \]
(FPCore (x) :precision binary64 (- (log (+ x 1.0)) (log x)))
↓
(FPCore (x) :precision binary64 (log1p (/ 1.0 x)))
double code(double x) {
	return log((x + 1.0)) - log(x);
}
↓
double code(double x) {
	return log1p((1.0 / x));
}

public static double code(double x) {
	return Math.log((x + 1.0)) - Math.log(x);
}
↓
public static double code(double x) {
	return Math.log1p((1.0 / x));
}

def code(x):
	return math.log((x + 1.0)) - math.log(x)
↓
def code(x):
	return math.log1p((1.0 / x))

function code(x)
	return Float64(log(Float64(x + 1.0)) - log(x))
end
↓
function code(x)
	return log1p(Float64(1.0 / x))
end

code[x_] := N[(N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]

\log \left(x + 1\right) - \log x
↓
\mathsf{log1p}\left(\frac{1}{x}\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 3.1

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

  2. Applied egg-rr2.2

    \[\leadsto \color{blue}{\log \left(\frac{x + 1}{x}\right)} \]

  3. Applied egg-rr2.3

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

  4. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)} \]
    Proof

    [Start]2.3

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

    *-lft-identity [<=]2.3

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

    associate-*r/ [=>]2.3

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

    associate-*l/ [<=]2.3

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

    distribute-rgt-in [=>]2.3

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

    +-commutative [=>]2.3

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

    rgt-mult-inverse [=>]2.2

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

    *-lft-identity [=>]2.2

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

    associate--l+ [=>]0.2

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

    metadata-eval [=>]0.2

    \[ \mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right) \]

  5. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(\frac{1}{x}\right) \]

Alternatives

Alternative 1
Error41.9
Cost6724
\[\begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;x - \log x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \frac{-0.5}{x \cdot x}\\ \end{array} \]
Alternative 2
Error2.2
Cost6720
\[\log \left(1 + \frac{1}{x}\right) \]
Alternative 3
Error43.7
Cost6660
\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-\log x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \frac{-0.5}{x \cdot x}\\ \end{array} \]
Alternative 4
Error45.7
Cost708
\[\begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \frac{-0.5}{x \cdot x}\\ \end{array} \]
Alternative 5
Error49.0
Cost192
\[\frac{1}{x} \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "log(x+1) - log(x)"
  :precision binary64
  :pre (and (<= 0.01 x) (<= x 1000.0))
  (- (log (+ x 1.0)) (log x)))