Average Error: 0.0 → 0.0
Time: 2.6s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[1 + \frac{-1}{\mathsf{fma}\left(\frac{2}{1 + t}, \sqrt[3]{\frac{8}{{\left(1 + t\right)}^{3}}} + -4, 6\right)} \]
(FPCore (t)
 :precision binary64
 (-
  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)))))))))
↓
(FPCore (t)
 :precision binary64
 (+
  1.0
  (/
   -1.0
   (fma (/ 2.0 (+ 1.0 t)) (+ (cbrt (/ 8.0 (pow (+ 1.0 t) 3.0))) -4.0) 6.0))))
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 + (-1.0 / fma((2.0 / (1.0 + t)), (cbrt((8.0 / pow((1.0 + t), 3.0))) + -4.0), 6.0));
}

function code(t)
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))))))
end
↓
function code(t)
	return Float64(1.0 + Float64(-1.0 / fma(Float64(2.0 / Float64(1.0 + t)), Float64(cbrt(Float64(8.0 / (Float64(1.0 + t) ^ 3.0))) + -4.0), 6.0)))
end

code[t_] := N[(1.0 - N[(1.0 / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[t_] := N[(1.0 + N[(-1.0 / N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(8.0 / N[Power[N[(1.0 + t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + -4.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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 + \frac{-1}{\mathsf{fma}\left(\frac{2}{1 + t}, \sqrt[3]{\frac{8}{{\left(1 + t\right)}^{3}}} + -4, 6\right)}

Error

Bits error versus t

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. Simplified0.0

    \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(\frac{2}{1 + t}, \frac{2}{1 + t} + -4, 6\right)}} \]

  3. Applied egg-rr0.0

    \[\leadsto 1 + \frac{-1}{\mathsf{fma}\left(\frac{2}{1 + t}, \color{blue}{\sqrt[3]{\frac{8}{{\left(1 + t\right)}^{3}}}} + -4, 6\right)} \]

  4. Final simplification0.0

    \[\leadsto 1 + \frac{-1}{\mathsf{fma}\left(\frac{2}{1 + t}, \sqrt[3]{\frac{8}{{\left(1 + t\right)}^{3}}} + -4, 6\right)} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 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)))))))))