Average Error: 0.0 → 0.0
Time: 10.0s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}, 2 - \frac{2}{1 + t}, 2\right)}\]
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}{\mathsf{fma}\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r535574 = 1.0;
        double r535575 = 2.0;
        double r535576 = t;
        double r535577 = r535575 / r535576;
        double r535578 = r535574 / r535576;
        double r535579 = r535574 + r535578;
        double r535580 = r535577 / r535579;
        double r535581 = r535575 - r535580;
        double r535582 = r535581 * r535581;
        double r535583 = r535575 + r535582;
        double r535584 = r535574 / r535583;
        double r535585 = r535574 - r535584;
        return r535585;
}
↓
double f(double t) {
        double r535586 = 1.0;
        double r535587 = 2.0;
        double r535588 = t;
        double r535589 = r535586 + r535588;
        double r535590 = r535587 / r535589;
        double r535591 = r535587 - r535590;
        double r535592 = cbrt(r535591);
        double r535593 = r535592 * r535592;
        double r535594 = r535593 * r535592;
        double r535595 = fma(r535594, r535591, r535587);
        double r535596 = r535586 / r535595;
        double r535597 = r535586 - r535596;
        return r535597;
}

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-cube-cbrt0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +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)))))))))