Average Error: 0.0 → 0.0
Time: 15.6s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)
double f(double t) {
        double r668068 = 1.0;
        double r668069 = 2.0;
        double r668070 = t;
        double r668071 = r668069 / r668070;
        double r668072 = r668068 / r668070;
        double r668073 = r668068 + r668072;
        double r668074 = r668071 / r668073;
        double r668075 = r668069 - r668074;
        double r668076 = r668075 * r668075;
        double r668077 = r668069 + r668076;
        double r668078 = r668068 / r668077;
        double r668079 = r668068 - r668078;
        return r668079;
}
↓
double f(double t) {
        double r668080 = 1.0;
        double r668081 = 2.0;
        double r668082 = t;
        double r668083 = r668080 + r668082;
        double r668084 = r668081 / r668083;
        double r668085 = r668081 - r668084;
        double r668086 = fma(r668085, r668085, r668081);
        double r668087 = r668080 / r668086;
        double r668088 = log1p(r668087);
        double r668089 = expm1(r668088);
        double r668090 = r668080 - r668089;
        return r668090;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}}\]
  3. Using strategy rm

  4. Applied expm1-log1p-u0.0

    \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)}\]

  5. Final simplification0.0

    \[\leadsto 1 - \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))