Average Error: 0.0 → 0.0
Time: 17.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\sqrt[3]{\left(\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right) \cdot \mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)\right) \cdot \mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}}\]
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}{\sqrt[3]{\left(\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right) \cdot \mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)\right) \cdot \mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}}
double f(double t) {
        double r1697810 = 1.0;
        double r1697811 = 2.0;
        double r1697812 = t;
        double r1697813 = r1697811 / r1697812;
        double r1697814 = r1697810 / r1697812;
        double r1697815 = r1697810 + r1697814;
        double r1697816 = r1697813 / r1697815;
        double r1697817 = r1697811 - r1697816;
        double r1697818 = r1697817 * r1697817;
        double r1697819 = r1697811 + r1697818;
        double r1697820 = r1697810 / r1697819;
        double r1697821 = r1697810 - r1697820;
        return r1697821;
}
↓
double f(double t) {
        double r1697822 = 1.0;
        double r1697823 = 2.0;
        double r1697824 = t;
        double r1697825 = r1697822 + r1697824;
        double r1697826 = r1697823 / r1697825;
        double r1697827 = r1697823 - r1697826;
        double r1697828 = fma(r1697827, r1697827, r1697823);
        double r1697829 = r1697828 * r1697828;
        double r1697830 = r1697829 * r1697828;
        double r1697831 = cbrt(r1697830);
        double r1697832 = r1697822 / r1697831;
        double r1697833 = r1697822 - r1697832;
        return r1697833;
}

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 add-cbrt-cube0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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)))))))))