Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 7.4s

Precision: binary64

Cost: 13952

Math

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

↓

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

FPCore

(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
  (cbrt
   (pow (/ 1.0 (- -6.0 (/ (+ (/ -4.0 (- -1.0 t)) -8.0) (+ 1.0 t)))) 3.0))))

C

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 + cbrt(pow((1.0 / (-6.0 - (((-4.0 / (-1.0 - t)) + -8.0) / (1.0 + t)))), 3.0));
}

Java

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 + Math.cbrt(Math.pow((1.0 / (-6.0 - (((-4.0 / (-1.0 - t)) + -8.0) / (1.0 + t)))), 3.0));
}

Julia

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 + cbrt((Float64(1.0 / Float64(-6.0 - Float64(Float64(Float64(-4.0 / Float64(-1.0 - t)) + -8.0) / Float64(1.0 + t)))) ^ 3.0)))
end

Wolfram

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[Power[N[Power[N[(1.0 / N[(-6.0 - N[(N[(N[(-4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

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. Simplified 0.0

   \[\leadsto \color{blue}{1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}} \]
   Proof

   [Start] 0.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   sub-neg [=>] 0.0	\[ \color{blue}{1 + \left(-\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right)} \]
   distribute-neg-frac [=>] 0.0	\[ 1 + \color{blue}{\frac{-1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]
   metadata-eval [=>] 0.0	\[ 1 + \frac{\color{blue}{-1}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2}} \]
   sub-neg [=>] 0.0	\[ 1 + \frac{-1}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + 2} \]
   distribute-lft-in [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + 2} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2\right)} + 2} \]
   *-commutative [<=] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{2 \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right) + 2} \]
   sub-neg [=>] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right) + 2} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot \color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2\right)}\right) + 2} \]
   distribute-rgt-in [=>] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2 + 2 \cdot 2\right)}\right) + 2} \]
   associate-+r+ [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2\right) + 2 \cdot 2\right)} + 2} \]
   associate-+l+ [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2\right) + \left(2 \cdot 2 + 2\right)}} \]
   cancel-sign-sub-inv [<=] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot 2\right)} + \left(2 \cdot 2 + 2\right)} \]
   *-commutative [=>] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - \color{blue}{2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right) + \left(2 \cdot 2 + 2\right)} \]
   cancel-sign-sub-inv [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(2 \cdot 2 + 2\right)} \]
   neg-mul-1 [=>] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [<=] 0.0	\[ 1 + \frac{-1}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\color{blue}{\left(-1\right)} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(2 \cdot 2 + 2\right)} \]
   associate-*r* [=>] 0.0	\[ 1 + \frac{-1}{\left(\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1\right)\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(2 \cdot 2 + 2\right)} \]
   distribute-rgt-out [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1\right) + \left(-2\right)\right)} + \left(2 \cdot 2 + 2\right)} \]
   +-commutative [<=] 0.0	\[ 1 + \frac{-1}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \color{blue}{\left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1\right)\right)} + \left(2 \cdot 2 + 2\right)} \]
   associate-/l/ [=>] 0.2	\[ 1 + \frac{-1}{\color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}} \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1\right)\right) + \left(2 \cdot 2 + 2\right)} \]
   associate-*l/ [=>] 0.2	\[ 1 + \frac{-1}{\color{blue}{\frac{2 \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1\right)\right)}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(2 \cdot 2 + 2\right)} \]
   distribute-rgt-in [=>] 0.2	\[ 1 + \frac{-1}{\frac{\color{blue}{\left(-2\right) \cdot 2 + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1\right)\right) \cdot 2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   associate-*l* [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot 2 + \color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(-1\right) \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\color{blue}{-1} \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{-2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [<=] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(-2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   *-commutative [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot 2 + \color{blue}{\left(-2\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   distribute-lft-in [<=] 0.2	\[ 1 + \frac{-1}{\frac{\color{blue}{\left(-2\right) \cdot \left(2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   sub-neg [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \left(2 + \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   associate-+r+ [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \color{blue}{\left(\left(2 + 2\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \left(\color{blue}{4} + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [<=] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \left(\color{blue}{2 \cdot 2} + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   +-commutative [<=] 0.2	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   distribute-lft-in [=>] 0.2	\[ 1 + \frac{-1}{\frac{\color{blue}{\left(-2\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-2\right) \cdot \left(2 \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   associate-/l/ [=>] 0.1	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \left(-\color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}}\right) + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   distribute-neg-frac [=>] 0.1	\[ 1 + \frac{-1}{\frac{\left(-2\right) \cdot \color{blue}{\frac{-2}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   associate-*r/ [=>] 0.1	\[ 1 + \frac{-1}{\frac{\color{blue}{\frac{\left(-2\right) \cdot \left(-2\right)}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.1	\[ 1 + \frac{-1}{\frac{\frac{\color{blue}{-2} \cdot \left(-2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.1	\[ 1 + \frac{-1}{\frac{\frac{-2 \cdot \color{blue}{-2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.1	\[ 1 + \frac{-1}{\frac{\frac{\color{blue}{4}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   *-commutative [=>] 0.1	\[ 1 + \frac{-1}{\frac{\frac{4}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   +-commutative [=>] 0.1	\[ 1 + \frac{-1}{\frac{\frac{4}{t \cdot \color{blue}{\left(\frac{1}{t} + 1\right)}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   distribute-lft-in [=>] 0.1	\[ 1 + \frac{-1}{\frac{\frac{4}{\color{blue}{t \cdot \frac{1}{t} + t \cdot 1}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   rgt-mult-inverse [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{\color{blue}{1} + t \cdot 1} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   *-rgt-identity [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + \color{blue}{t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + \color{blue}{-2} \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -2 \cdot \color{blue}{4}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + \color{blue}{-8}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 \cdot 2 + 2\right)} \]
   *-commutative [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}} + \left(2 \cdot 2 + 2\right)} \]
   +-commutative [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{t \cdot \color{blue}{\left(\frac{1}{t} + 1\right)}} + \left(2 \cdot 2 + 2\right)} \]
   distribute-lft-in [=>] 0.2	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{\color{blue}{t \cdot \frac{1}{t} + t \cdot 1}} + \left(2 \cdot 2 + 2\right)} \]
   rgt-mult-inverse [=>] 0.0	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{\color{blue}{1} + t \cdot 1} + \left(2 \cdot 2 + 2\right)} \]
   *-rgt-identity [=>] 0.0	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + \color{blue}{t}} + \left(2 \cdot 2 + 2\right)} \]
   metadata-eval [=>] 0.0	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + \left(\color{blue}{4} + 2\right)} \]
   metadata-eval [=>] 0.0	\[ 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + \color{blue}{6}} \]

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;1 + \left(t \cdot t + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.037037037037037035}{t \cdot t} + \left(-0.16666666666666666 - \frac{0.2222222222222222}{t}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.0
Cost	1088

\[1 + \frac{-1}{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 6} \]

Alternative 3
Error	0.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t \cdot t + -0.5\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5
Error	0.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.335:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 6
Error	26.0
Cost	64

\[0.8333333333333334 \]

Reproduce

herbie shell --seed 2022356 
(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)))))))))