Average Error: 0.0 → 0.0
Time: 6.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}\]
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(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r429217 = 1.0;
        double r429218 = 2.0;
        double r429219 = t;
        double r429220 = r429218 / r429219;
        double r429221 = r429217 / r429219;
        double r429222 = r429217 + r429221;
        double r429223 = r429220 / r429222;
        double r429224 = r429218 - r429223;
        double r429225 = r429224 * r429224;
        double r429226 = r429218 + r429225;
        double r429227 = r429217 / r429226;
        double r429228 = r429217 - r429227;
        return r429228;
}
↓
double f(double t) {
        double r429229 = 1.0;
        double r429230 = 2.0;
        double r429231 = t;
        double r429232 = r429229 + r429231;
        double r429233 = r429230 / r429232;
        double r429234 = r429230 - r429233;
        double r429235 = fma(r429234, r429234, r429230);
        double r429236 = r429229 / r429235;
        double r429237 = r429229 - r429236;
        return r429237;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019155 +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)))))))))