ln(1 + x)

Average Error: 39.1 → 0.4

Time: 11.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.000000000007547740210611664224416017532:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{x}{1}, \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

C

double f(double x) {
        double r3341976 = 1.0;
        double r3341977 = x;
        double r3341978 = r3341976 + r3341977;
        double r3341979 = log(r3341978);
        return r3341979;
}

↓

double f(double x) {
        double r3341980 = x;
        double r3341981 = 1.0;
        double r3341982 = r3341980 + r3341981;
        double r3341983 = 1.0000000000075477;
        bool r3341984 = r3341982 <= r3341983;
        double r3341985 = -0.5;
        double r3341986 = r3341980 / r3341981;
        double r3341987 = r3341985 * r3341986;
        double r3341988 = log(r3341981);
        double r3341989 = fma(r3341981, r3341980, r3341988);
        double r3341990 = fma(r3341987, r3341986, r3341989);
        double r3341991 = log(r3341982);
        double r3341992 = r3341984 ? r3341990 : r3341991;
        return r3341992;
}

Error

Bits error versus x

Target

Original	39.1
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (+ 1.0 x) < 1.0000000000075477

   1. Initial program 59.4

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

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\log 1 + 1 \cdot x\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{x}{1}, \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)}\]

   if 1.0000000000075477 < (+ 1.0 x)

   1. Initial program 0.5

      \[\log \left(1 + x\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.000000000007547740210611664224416017532:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{x}{1}, \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))