Average Error: 0.1 → 0.1
Time: 4.3s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \log \left(e^{{\left(\frac{2 \cdot t}{1 + t}\right)}^{2}}\right)}\]
\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{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \log \left(e^{{\left(\frac{2 \cdot t}{1 + t}\right)}^{2}}\right)}
double code(double t) {
	return ((1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))));
}
↓
double code(double t) {
	return ((1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + log(exp(pow(((2.0 * t) / (1.0 + t)), 2.0)))));
}

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

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

  4. Simplified16.4

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

  6. Applied pow1/216.4

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

  7. Applied pow-pow0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 
(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))))))