Average Error: 0.1 → 0.1
Time: 4.8s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{{1}^{3} + {\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}^{3}}{\left(\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t} - 1\right) + 1 \cdot 1\right) \cdot \left(2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}\]
\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}^{3} + {\left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}^{3}}{\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} - 1\right) + 1 \cdot 1\right) \cdot \left(2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}
double f(double t) {
        double r54329 = 1.0;
        double r54330 = 2.0;
        double r54331 = t;
        double r54332 = r54330 * r54331;
        double r54333 = r54329 + r54331;
        double r54334 = r54332 / r54333;
        double r54335 = r54334 * r54334;
        double r54336 = r54329 + r54335;
        double r54337 = r54330 + r54335;
        double r54338 = r54336 / r54337;
        return r54338;
}
↓
double f(double t) {
        double r54339 = 1.0;
        double r54340 = 3.0;
        double r54341 = pow(r54339, r54340);
        double r54342 = 2.0;
        double r54343 = t;
        double r54344 = r54342 * r54343;
        double r54345 = r54339 + r54343;
        double r54346 = r54344 / r54345;
        double r54347 = r54346 * r54346;
        double r54348 = pow(r54347, r54340);
        double r54349 = r54341 + r54348;
        double r54350 = r54347 - r54339;
        double r54351 = r54347 * r54350;
        double r54352 = r54339 * r54339;
        double r54353 = r54351 + r54352;
        double r54354 = r54342 + r54347;
        double r54355 = r54353 * r54354;
        double r54356 = r54349 / r54355;
        return r54356;
}

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 flip3-+0.1

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

  4. Applied associate-/l/0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

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))))))