Average Error: 0.0 → 0.0
Time: 15.4s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\frac{\left(2 \cdot 2\right) \cdot 2 - \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t}\right) \cdot \frac{2}{1 + t}}{\mathsf{fma}\left(\frac{2}{1 + t}, 2 + \frac{2}{1 + t}, 2 \cdot 2\right)}, 2 - \frac{2}{1 + t}, 2\right)}\]
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(\frac{\left(2 \cdot 2\right) \cdot 2 - \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t}\right) \cdot \frac{2}{1 + t}}{\mathsf{fma}\left(\frac{2}{1 + t}, 2 + \frac{2}{1 + t}, 2 \cdot 2\right)}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r1031568 = 1.0;
        double r1031569 = 2.0;
        double r1031570 = t;
        double r1031571 = r1031569 / r1031570;
        double r1031572 = r1031568 / r1031570;
        double r1031573 = r1031568 + r1031572;
        double r1031574 = r1031571 / r1031573;
        double r1031575 = r1031569 - r1031574;
        double r1031576 = r1031575 * r1031575;
        double r1031577 = r1031569 + r1031576;
        double r1031578 = r1031568 / r1031577;
        double r1031579 = r1031568 - r1031578;
        return r1031579;
}
↓
double f(double t) {
        double r1031580 = 1.0;
        double r1031581 = 2.0;
        double r1031582 = r1031581 * r1031581;
        double r1031583 = r1031582 * r1031581;
        double r1031584 = t;
        double r1031585 = r1031580 + r1031584;
        double r1031586 = r1031581 / r1031585;
        double r1031587 = r1031586 * r1031586;
        double r1031588 = r1031587 * r1031586;
        double r1031589 = r1031583 - r1031588;
        double r1031590 = r1031581 + r1031586;
        double r1031591 = fma(r1031586, r1031590, r1031582);
        double r1031592 = r1031589 / r1031591;
        double r1031593 = r1031581 - r1031586;
        double r1031594 = fma(r1031592, r1031593, r1031581);
        double r1031595 = r1031580 / r1031594;
        double r1031596 = r1031580 - r1031595;
        return r1031596;
}

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

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

  4. Applied flip3--0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{{2}^{3} - {\left(\frac{2}{1 + t}\right)}^{3}}{2 \cdot 2 + \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t} + 2 \cdot \frac{2}{1 + t}\right)}}, 2 - \frac{2}{1 + t}, 2\right)}\]

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +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)))))))))