Average Error: 0.0 → 0.0
Time: 18.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\right)\right)\]
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(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\right)\right)
double f(double t) {
        double r26507 = 1.0;
        double r26508 = 2.0;
        double r26509 = t;
        double r26510 = r26508 / r26509;
        double r26511 = r26507 / r26509;
        double r26512 = r26507 + r26511;
        double r26513 = r26510 / r26512;
        double r26514 = r26508 - r26513;
        double r26515 = r26514 * r26514;
        double r26516 = r26508 + r26515;
        double r26517 = r26507 / r26516;
        double r26518 = r26507 - r26517;
        return r26518;
}
↓
double f(double t) {
        double r26519 = 1.0;
        double r26520 = 2.0;
        double r26521 = t;
        double r26522 = fma(r26519, r26521, r26519);
        double r26523 = r26520 / r26522;
        double r26524 = r26520 - r26523;
        double r26525 = fma(r26524, r26524, r26520);
        double r26526 = r26519 / r26525;
        double r26527 = log1p(r26526);
        double r26528 = expm1(r26527);
        double r26529 = r26519 - r26528;
        return r26529;
}

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

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

  4. Applied expm1-log1p-u0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))