Average Error: 0.1 → 0.1
Time: 10.6s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\log \left(e^{\frac{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t} + 1}{{\left(\frac{2 \cdot t}{1 + t}\right)}^{6} + {2}^{3}} \cdot \left(2 \cdot 2 + \left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) - 2 \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)\right)}\right)\]
\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}}
↓
\log \left(e^{\frac{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t} + 1}{{\left(\frac{2 \cdot t}{1 + t}\right)}^{6} + {2}^{3}} \cdot \left(2 \cdot 2 + \left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) - 2 \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)\right)}\right)
double f(double t) {
        double r73634 = 1.0;
        double r73635 = 2.0;
        double r73636 = t;
        double r73637 = r73635 * r73636;
        double r73638 = r73634 + r73636;
        double r73639 = r73637 / r73638;
        double r73640 = r73639 * r73639;
        double r73641 = r73634 + r73640;
        double r73642 = r73635 + r73640;
        double r73643 = r73641 / r73642;
        return r73643;
}
↓
double f(double t) {
        double r73644 = 2.0;
        double r73645 = t;
        double r73646 = r73644 * r73645;
        double r73647 = 1.0;
        double r73648 = r73647 + r73645;
        double r73649 = r73646 / r73648;
        double r73650 = r73649 * r73649;
        double r73651 = r73650 + r73647;
        double r73652 = 6.0;
        double r73653 = pow(r73649, r73652);
        double r73654 = 3.0;
        double r73655 = pow(r73644, r73654);
        double r73656 = r73653 + r73655;
        double r73657 = r73651 / r73656;
        double r73658 = r73644 * r73644;
        double r73659 = r73650 * r73650;
        double r73660 = r73644 * r73650;
        double r73661 = r73659 - r73660;
        double r73662 = r73658 + r73661;
        double r73663 = r73657 * r73662;
        double r73664 = exp(r73663);
        double r73665 = log(r73664);
        return r73665;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
  2. Using strategy rm

  3. Applied add-log-exp0.1

    \[\leadsto \color{blue}{\log \left(e^{\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}}\right)}\]
  4. Using strategy rm

  5. Applied flip3-+0.1

    \[\leadsto \log \left(e^{\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{\frac{{2}^{3} + {\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}^{3}}{2 \cdot 2 + \left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) - 2 \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)}}}}\right)\]

  6. Applied associate-/r/0.1

    \[\leadsto \log \left(e^{\color{blue}{\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{{2}^{3} + {\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}^{3}} \cdot \left(2 \cdot 2 + \left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) - 2 \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)\right)}}\right)\]

  7. Simplified0.1

    \[\leadsto \log \left(e^{\color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t} + 1}{{\left(\frac{2 \cdot t}{1 + t}\right)}^{6} + {2}^{3}}} \cdot \left(2 \cdot 2 + \left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) - 2 \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)\right)}\right)\]

  8. Final simplification0.1

    \[\leadsto \log \left(e^{\frac{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t} + 1}{{\left(\frac{2 \cdot t}{1 + t}\right)}^{6} + {2}^{3}} \cdot \left(2 \cdot 2 + \left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) - 2 \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)\right)}\right)\]

Reproduce

herbie shell --seed 2020046 
(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))))))