Average Error: 39.1 → 0.0
Time: 3.6s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\log_* (1 + x)\]
double f(double x) {
        double r3942522 = 1.0;
        double r3942523 = x;
        double r3942524 = r3942522 + r3942523;
        double r3942525 = log(r3942524);
        return r3942525;
}
↓
double f(double x) {
        double r3942526 = x;
        double r3942527 = log1p(r3942526);
        return r3942527;
}


\log \left(1 + x\right)
↓
\log_* (1 + x)

Error

Bits error versus x

Target

Original39.1
Target0.2
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Initial program 39.1

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\log_* (1 + x)}\]

  3. Final simplification0.0

    \[\leadsto \log_* (1 + x)\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))