Average Error: 0.0 → 0.0
Time: 31.0s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{\sqrt{2}}{\frac{1 + t}{\sqrt{2}}}, 2\right)}\]
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{\sqrt{2}}{\frac{1 + t}{\sqrt{2}}}, 2\right)}
double f(double t) {
        double r977169 = 1.0;
        double r977170 = 2.0;
        double r977171 = t;
        double r977172 = r977170 / r977171;
        double r977173 = r977169 / r977171;
        double r977174 = r977169 + r977173;
        double r977175 = r977172 / r977174;
        double r977176 = r977170 - r977175;
        double r977177 = r977176 * r977176;
        double r977178 = r977170 + r977177;
        double r977179 = r977169 / r977178;
        double r977180 = r977169 - r977179;
        return r977180;
}
↓
double f(double t) {
        double r977181 = 1.0;
        double r977182 = 2.0;
        double r977183 = t;
        double r977184 = r977181 + r977183;
        double r977185 = r977182 / r977184;
        double r977186 = r977182 - r977185;
        double r977187 = sqrt(r977182);
        double r977188 = r977184 / r977187;
        double r977189 = r977187 / r977188;
        double r977190 = r977182 - r977189;
        double r977191 = fma(r977186, r977190, r977182);
        double r977192 = r977181 / r977191;
        double r977193 = r977181 - r977192;
        return r977193;
}

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-sqr-sqrt0.0

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

  5. Applied associate-/l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 +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)))))))))