Average Error: 0.0 → 0.0
Time: 29.0s
Precision: binary64
Cost: 1472
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[1 - \frac{1}{2 + \frac{4 + \frac{\frac{4}{t}}{\frac{-1 - t}{t}}}{\frac{1 + t}{t}}} \]
(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 (+ 2.0 (/ (+ 4.0 (/ (/ 4.0 t) (/ (- -1.0 t) t))) (/ (+ 1.0 t) t))))))
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 / (2.0 + ((4.0 + ((4.0 / t) / ((-1.0 - t) / t))) / ((1.0 + t) / t))));
}

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 - (1.0d0 / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))))))
end function
↓
real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 - (1.0d0 / (2.0d0 + ((4.0d0 + ((4.0d0 / t) / (((-1.0d0) - t) / t))) / ((1.0d0 + t) / t))))
end function

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 1.0 - (1.0 / (2.0 + ((4.0 + ((4.0 / t) / ((-1.0 - t) / t))) / ((1.0 + t) / t))));
}

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 1.0 - (1.0 / (2.0 + ((4.0 + ((4.0 / t) / ((-1.0 - t) / t))) / ((1.0 + t) / t))))

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 / Float64(2.0 + Float64(Float64(4.0 + Float64(Float64(4.0 / t) / Float64(Float64(-1.0 - t) / t))) / Float64(Float64(1.0 + t) / t)))))
end

function tmp = code(t)
	tmp = 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)))))));
end
↓
function tmp = code(t)
	tmp = 1.0 - (1.0 / (2.0 + ((4.0 + ((4.0 / t) / ((-1.0 - t) / t))) / ((1.0 + t) / t))));
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[(2.0 + N[(N[(4.0 + N[(N[(4.0 / t), $MachinePrecision] / N[(N[(-1.0 - t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t), $MachinePrecision] / t), $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)}
↓
1 - \frac{1}{2 + \frac{4 + \frac{\frac{4}{t}}{\frac{-1 - t}{t}}}{\frac{1 + t}{t}}}

Error

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

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

  3. Taylor expanded in t around 0 0.0

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

  4. Applied egg-rr0.0

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

  5. Simplified0.0

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{4 + \frac{\frac{4}{t}}{\frac{-1 - t}{t}}}{\frac{1 + t}{t}}}} \]
    Proof

Alternatives

Alternative 1
Error0.6
Cost1736
\[\begin{array}{l} \mathbf{if}\;t \leq -1.55:\\ \;\;\;\;\frac{-1}{6 + \frac{-8}{t}} - -1\\ \mathbf{elif}\;t \leq 1.25:\\ \;\;\;\;1 - \frac{1}{2 + \frac{t \cdot -4}{\frac{-1 - t}{t}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{t}\right)}\\ \end{array} \]
Alternative 2
Error0.6
Cost1352
\[\begin{array}{l} t_1 := 2 - \frac{2}{t}\\ \mathbf{if}\;t \leq -1.55:\\ \;\;\;\;\frac{-1}{6 + \frac{-8}{t}} - -1\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;1 - \frac{1}{2 + \frac{t \cdot -4}{\frac{-1 - t}{t}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{2 + t_1 \cdot t_1}\\ \end{array} \]
Alternative 3
Error0.6
Cost1352
\[\begin{array}{l} \mathbf{if}\;t \leq -1.55:\\ \;\;\;\;\frac{-1}{6 + \frac{-8}{t}} - -1\\ \mathbf{elif}\;t \leq 1.25:\\ \;\;\;\;1 - \frac{1}{2 + \frac{t \cdot -4}{\frac{-1 - t}{t}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{2 + \frac{4 - \frac{4}{t}}{\frac{1 + t}{t}}}\\ \end{array} \]
Alternative 4
Error0.6
Cost1224
\[\begin{array}{l} t_1 := \frac{-1}{6 + \frac{-8}{t}} - -1\\ \mathbf{if}\;t \leq -1.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.35:\\ \;\;\;\;1 - \frac{1}{2 + \frac{t \cdot -4}{\frac{-1 - t}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error0.6
Cost968
\[\begin{array}{l} t_1 := \frac{-1}{6 + \frac{-8}{t}} - -1\\ \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1:\\ \;\;\;\;1 - \frac{1}{2 + \left(t \cdot t\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error0.6
Cost840
\[\begin{array}{l} t_1 := \frac{-1}{6 + \frac{-8}{t}} - -1\\ \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.66:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error0.9
Cost584
\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 8
Error1.0
Cost456
\[\begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;1 - 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 9
Error26.5
Cost64
\[0.8333333333333334 \]

Error

Reproduce

herbie shell --seed 2023010 
(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)))))))))