Average Error: 0.0 → 0.0
Time: 32.2s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(\log \left(e^{\frac{t \cdot 2}{1 + t}}\right)\right), \left(\frac{t \cdot 2}{1 + t}\right), 2\right)}\]
\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(\left(\frac{t \cdot 2}{1 + t}\right), \left(\frac{t \cdot 2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(\log \left(e^{\frac{t \cdot 2}{1 + t}}\right)\right), \left(\frac{t \cdot 2}{1 + t}\right), 2\right)}
double f(double t) {
        double r17325467 = 1.0;
        double r17325468 = 2.0;
        double r17325469 = t;
        double r17325470 = r17325468 * r17325469;
        double r17325471 = r17325467 + r17325469;
        double r17325472 = r17325470 / r17325471;
        double r17325473 = r17325472 * r17325472;
        double r17325474 = r17325467 + r17325473;
        double r17325475 = r17325468 + r17325473;
        double r17325476 = r17325474 / r17325475;
        return r17325476;
}
↓
double f(double t) {
        double r17325477 = t;
        double r17325478 = 2.0;
        double r17325479 = r17325477 * r17325478;
        double r17325480 = 1.0;
        double r17325481 = r17325480 + r17325477;
        double r17325482 = r17325479 / r17325481;
        double r17325483 = fma(r17325482, r17325482, r17325480);
        double r17325484 = exp(r17325482);
        double r17325485 = log(r17325484);
        double r17325486 = fma(r17325485, r17325482, r17325478);
        double r17325487 = r17325483 / r17325486;
        return r17325487;
}

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

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

  4. Applied add-log-exp0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))