Average Error: 0.0 → 0.0
Time: 23.2s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, 2 \cdot \frac{t}{1 + t}, 2\right)}\]
\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\frac{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, \frac{2 \cdot t}{1 + t}, 1\right)}{\mathsf{fma}\left(\frac{2 \cdot t}{1 + t}, 2 \cdot \frac{t}{1 + t}, 2\right)}
double f(double t) {
        double r50951 = 1.0;
        double r50952 = 2.0;
        double r50953 = t;
        double r50954 = r50952 * r50953;
        double r50955 = r50951 + r50953;
        double r50956 = r50954 / r50955;
        double r50957 = r50956 * r50956;
        double r50958 = r50951 + r50957;
        double r50959 = r50952 + r50957;
        double r50960 = r50958 / r50959;
        return r50960;
}
↓
double f(double t) {
        double r50961 = 2.0;
        double r50962 = t;
        double r50963 = r50961 * r50962;
        double r50964 = 1.0;
        double r50965 = r50964 + r50962;
        double r50966 = r50963 / r50965;
        double r50967 = fma(r50966, r50966, r50964);
        double r50968 = r50962 / r50965;
        double r50969 = r50961 * r50968;
        double r50970 = fma(r50966, r50969, r50961);
        double r50971 = r50967 / r50970;
        return r50971;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Simplified0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))