Average Error: 0.0 → 0.0
Time: 8.0s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(1, 2, -\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}, \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}, \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)\right)}\right)\right)\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(1, 2, -\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}, \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}, \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)\right)}\right)\right)
double code(double t) {
	return (1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))))));
}
↓
double code(double t) {
	return (1.0 - log1p(expm1((1.0 / (2.0 + (((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * fma(1.0, 2.0, -((cbrt((2.0 / t)) / (1.0 + (1.0 / t))) * ((cbrt((2.0 / t)) * cbrt((2.0 / t))) / 1.0)))) + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * fma(-(cbrt((2.0 / t)) / (1.0 + (1.0 / t))), ((cbrt((2.0 / t)) * cbrt((2.0 / t))) / 1.0), ((cbrt((2.0 / t)) / (1.0 + (1.0 / t))) * ((cbrt((2.0 / t)) * cbrt((2.0 / t))) / 1.0))))))))));
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
  2. Using strategy rm

  3. Applied log1p-expm1-u0.0

    \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right)\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{\color{blue}{1 \cdot \left(1 + \frac{1}{t}\right)}}\right)}\right)\right)\]

  6. Applied add-cube-cbrt0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \sqrt[3]{\frac{2}{t}}}}{1 \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)\right)\]

  7. Applied times-frac0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}}\right)}\right)\right)\]

  8. Applied add-sqr-sqrt0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\color{blue}{\sqrt{2} \cdot \sqrt{2}} - \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}\right)}\right)\right)\]

  9. Applied prod-diff0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{2}, -\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}, \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}, \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)\right)}}\right)\right)\]

  10. Applied distribute-lft-in0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{2}, -\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}, \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}, \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)\right)}}\right)\right)\]

  11. Simplified0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(1, 2, -\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)} + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}, \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}, \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)\right)}\right)\right)\]

  12. Final simplification0.0

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(1, 2, -\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}, \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}, \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}\right)\right)}\right)\right)\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))