Average Error: 0.0 → 0.0
Time: 18.2s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]
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 - \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)}
double f(double t) {
        double r28570 = 1.0;
        double r28571 = 2.0;
        double r28572 = t;
        double r28573 = r28571 / r28572;
        double r28574 = r28570 / r28572;
        double r28575 = r28570 + r28574;
        double r28576 = r28573 / r28575;
        double r28577 = r28571 - r28576;
        double r28578 = r28577 * r28577;
        double r28579 = r28571 + r28578;
        double r28580 = r28570 / r28579;
        double r28581 = r28570 - r28580;
        return r28581;
}
↓
double f(double t) {
        double r28582 = 1.0;
        double r28583 = 2.0;
        double r28584 = t;
        double r28585 = fma(r28582, r28584, r28582);
        double r28586 = r28583 / r28585;
        double r28587 = r28583 - r28586;
        double r28588 = fma(r28587, r28587, r28583);
        double r28589 = r28582 / r28588;
        double r28590 = r28582 - r28589;
        return r28590;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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)))))))))