Average Error: 0.1 → 0.1
Time: 3.1s
Precision: binary64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \sqrt[3]{{\left(t \cdot \frac{2}{1 + t}\right)}^{6}}}\]
\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 + \sqrt[3]{{\left(t \cdot \frac{2}{1 + t}\right)}^{6}}}
(FPCore (t)
 :precision binary64
 (/
  (+ 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))))))
↓
(FPCore (t)
 :precision binary64
 (/
  (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))
  (+ 2.0 (cbrt (pow (* t (/ 2.0 (+ 1.0 t))) 6.0)))))
double code(double t) {
	return (((double) (1.0 + ((double) ((((double) (2.0 * t)) / ((double) (1.0 + t))) * (((double) (2.0 * t)) / ((double) (1.0 + t))))))) / ((double) (2.0 + ((double) ((((double) (2.0 * t)) / ((double) (1.0 + t))) * (((double) (2.0 * t)) / ((double) (1.0 + t))))))));
}
↓
double code(double t) {
	return (((double) (1.0 + ((double) ((((double) (2.0 * t)) / ((double) (1.0 + t))) * (((double) (2.0 * t)) / ((double) (1.0 + t))))))) / ((double) (2.0 + ((double) cbrt(((double) pow(((double) (t * (2.0 / ((double) (1.0 + t))))), 6.0)))))));
}

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-cbrt-cube17.3

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

  4. Applied add-cbrt-cube21.0

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

  5. Applied add-cbrt-cube21.0

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

  6. Applied cbrt-unprod21.0

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

  7. Applied cbrt-undiv21.0

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

  8. Applied add-cbrt-cube21.0

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

  9. Applied add-cbrt-cube21.0

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

  10. Applied add-cbrt-cube21.0

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

  11. Applied cbrt-unprod21.0

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

  12. Applied cbrt-undiv21.0

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

  13. Applied cbrt-unprod21.0

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020198 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 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))))))