Average Error: 0.0 → 0.0
Time: 2.7s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 + \frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{2}}\right)} \]
(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
 (exp
  (log1p (/ -1.0 (+ 2.0 (pow (+ 2.0 (/ -2.0 (* t (+ 1.0 (/ 1.0 t))))) 2.0))))))
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 exp(log1p((-1.0 / (2.0 + pow((2.0 + (-2.0 / (t * (1.0 + (1.0 / t))))), 2.0)))));
}

public static 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)))))));
}
↓
public static double code(double t) {
	return Math.exp(Math.log1p((-1.0 / (2.0 + Math.pow((2.0 + (-2.0 / (t * (1.0 + (1.0 / t))))), 2.0)))));
}

def code(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)))))))
↓
def code(t):
	return math.exp(math.log1p((-1.0 / (2.0 + math.pow((2.0 + (-2.0 / (t * (1.0 + (1.0 / t))))), 2.0)))))

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 exp(log1p(Float64(-1.0 / Float64(2.0 + (Float64(2.0 + Float64(-2.0 / Float64(t * Float64(1.0 + Float64(1.0 / t))))) ^ 2.0)))))
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[Exp[N[Log[1 + N[(-1.0 / N[(2.0 + N[Power[N[(2.0 + N[(-2.0 / N[(t * N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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)}
↓
e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 + \frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{2}}\right)}

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. Applied egg-rr0.0

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

  3. Final simplification0.0

    \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 + \frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{2}}\right)} \]

Reproduce

herbie shell --seed 2022166 
(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)))))))))