?

Average Error: 2.9 → 0.4
Time: 20.8s
Precision: binary64
Cost: 19648

?

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

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

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

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

Error?

Derivation?

  1. Initial program 2.9

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

  2. Simplified2.4

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

    [Start]2.9

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

    associate--l- [=>]2.9

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

    *-commutative [=>]2.9

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

    +-commutative [=>]2.9

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

    distribute-lft-in [=>]2.8

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

    associate--l+ [=>]2.8

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

    +-commutative [=>]2.8

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

    associate-+l- [=>]2.8

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

    associate--l+ [=>]2.8

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

    associate--l- [<=]2.8

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

    *-lft-identity [<=]2.8

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

    *-commutative [<=]2.8

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

    *-commutative [=>]2.8

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

    distribute-rgt-out-- [=>]2.6

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

    sub-neg [=>]2.6

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

  3. Applied egg-rr1.6

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

  4. Applied egg-rr1.6

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

  5. Simplified0.4

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

    [Start]1.6

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

    *-lft-identity [<=]1.6

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

    associate-*l/ [<=]1.7

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

    distribute-rgt-in [=>]1.7

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

    rgt-mult-inverse [=>]1.6

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

    *-lft-identity [=>]1.6

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

    +-commutative [=>]1.6

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

    associate--l+ [=>]0.4

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

    metadata-eval [=>]0.4

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

  6. Final simplification0.4

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

Alternatives

Alternative 1
Error2.4
Cost19776
\[\left(-1 + \mathsf{log1p}\left(x\right)\right) + x \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right) \]
Alternative 2
Error1.6
Cost19776
\[\mathsf{fma}\left(x, \log \left(1 + \frac{1}{x}\right), -1 + \mathsf{log1p}\left(x\right)\right) \]
Alternative 3
Error37.8
Cost13508
\[\begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x + 1}{x}\right), x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log x + \left(\frac{0.5}{x} + \frac{-0.16666666666666666}{x \cdot x}\right)\\ \end{array} \]
Alternative 4
Error2.7
Cost13504
\[\mathsf{log1p}\left(x\right) \cdot \left(x + 1\right) + \left(-1 - x \cdot \log x\right) \]
Alternative 5
Error2.7
Cost13504
\[\left(-1 + \mathsf{log1p}\left(x\right) \cdot \left(x + 1\right)\right) - x \cdot \log x \]
Alternative 6
Error37.9
Cost7236
\[\begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;-1 + \left(x - x \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\log x + \left(\frac{0.5}{x} + \frac{-0.16666666666666666}{x \cdot x}\right)\\ \end{array} \]
Alternative 7
Error38.0
Cost7236
\[\begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;-1 + \left(x \cdot \left(x + 1\right) - x \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\log x + \left(\frac{0.5}{x} + \frac{-0.16666666666666666}{x \cdot x}\right)\\ \end{array} \]
Alternative 8
Error41.3
Cost7108
\[\begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;-1 + \left(x - x \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \left(\log x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
Alternative 9
Error43.4
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;-1 + \mathsf{log1p}\left(x\right) \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log x + \frac{0.5}{x}\\ \end{array} \]
Alternative 10
Error41.3
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;-1 + x \cdot \left(1 - \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\log x + \frac{0.5}{x}\\ \end{array} \]
Alternative 11
Error41.3
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;-1 + \left(x - x \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\log x + \frac{0.5}{x}\\ \end{array} \]
Alternative 12
Error43.5
Cost6852
\[\begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\log x + \frac{0.5}{x}\\ \end{array} \]
Alternative 13
Error48.8
Cost6656
\[-\log \left(\frac{1}{x}\right) \]
Alternative 14
Error48.8
Cost6464
\[\log x \]
Alternative 15
Error58.9
Cost64
\[-1 \]

Error

Reproduce?

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