Kahan p13 Example 3

?

Percentage Accurate: 100.0% → 100.0%
Time: 35.6s
Precision: binary64
Cost: 8320

?

\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[1 + \frac{-1}{2 + \frac{2 + \frac{-2}{1 + t}}{\frac{2 + \frac{2}{1 + t}}{4 + -4 \cdot {\left(1 + t\right)}^{-2}}}} \]
(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
    (/
     (+ 2.0 (/ -2.0 (+ 1.0 t)))
     (/ (+ 2.0 (/ 2.0 (+ 1.0 t))) (+ 4.0 (* -4.0 (pow (+ 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 1.0 + (-1.0 / (2.0 + ((2.0 + (-2.0 / (1.0 + t))) / ((2.0 + (2.0 / (1.0 + t))) / (4.0 + (-4.0 * pow((1.0 + t), -2.0)))))));
}

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 + ((2.0d0 + ((-2.0d0) / (1.0d0 + t))) / ((2.0d0 + (2.0d0 / (1.0d0 + t))) / (4.0d0 + ((-4.0d0) * ((1.0d0 + t) ** (-2.0d0))))))))
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 + ((2.0 + (-2.0 / (1.0 + t))) / ((2.0 + (2.0 / (1.0 + t))) / (4.0 + (-4.0 * Math.pow((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 1.0 + (-1.0 / (2.0 + ((2.0 + (-2.0 / (1.0 + t))) / ((2.0 + (2.0 / (1.0 + t))) / (4.0 + (-4.0 * math.pow((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 Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))) / Float64(Float64(2.0 + Float64(2.0 / Float64(1.0 + t))) / Float64(4.0 + Float64(-4.0 * (Float64(1.0 + t) ^ -2.0))))))))
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 + ((2.0 + (-2.0 / (1.0 + t))) / ((2.0 + (2.0 / (1.0 + t))) / (4.0 + (-4.0 * ((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[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(-4.0 * N[Power[N[(1.0 + t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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{2 + \frac{-2}{1 + t}}{\frac{2 + \frac{2}{1 + t}}{4 + -4 \cdot {\left(1 + t\right)}^{-2}}}}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 10 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 100.0%

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

  2. Applied egg-rr100.0%

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}} \]
    Step-by-step derivation

    [Start]100.0%

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

    flip-- [=>]100.0%

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

    associate-*r/ [=>]100.0%

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

    metadata-eval [=>]100.0%

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

    pow2 [=>]100.0%

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

  3. Simplified100.0%

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{2 - \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}}{\frac{2 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}}{4 - {\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{2}}}}} \]
    Step-by-step derivation

    [Start]100.0%

    \[ 1 - \frac{1}{2 + \frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}} \]

    associate-/l* [=>]100.0%

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

    associate-/r* [<=]100.0%

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

    associate-/r* [<=]100.0%

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

    associate-/r* [<=]100.0%

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

  4. Applied egg-rr100.0%

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{1}{2 + \frac{2}{t + 1}} \cdot \frac{2 - \frac{2}{t + 1}}{\frac{1}{4 - \frac{4}{{\left(t + 1\right)}^{2}}}}}} \]
    Step-by-step derivation

    [Start]100.0%

    \[ 1 - \frac{1}{2 + \frac{2 - \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}}{\frac{2 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}}{4 - {\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{2}}}} \]

    *-un-lft-identity [=>]100.0%

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

    div-inv [=>]100.0%

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

    times-frac [=>]100.0%

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

    distribute-rgt-in [=>]100.0%

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

    *-un-lft-identity [<=]100.0%

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

    inv-pow [=>]100.0%

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

    pow-plus [=>]100.0%

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

    metadata-eval [=>]100.0%

    \[ 1 - \frac{1}{2 + \frac{1}{2 + \frac{2}{t + {t}^{\color{blue}{0}}}} \cdot \frac{2 - \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}}{\frac{1}{4 - {\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{2}}}} \]

    metadata-eval [=>]100.0%

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

  5. Applied egg-rr100.0%

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(2 - \frac{2}{t + 1}\right) \cdot \left(4 - 4 \cdot {\left(t + 1\right)}^{-2}\right)}{2 + \frac{2}{t + 1}}\right)} - 1\right)}} \]
    Step-by-step derivation

    [Start]100.0%

    \[ 1 - \frac{1}{2 + \frac{1}{2 + \frac{2}{t + 1}} \cdot \frac{2 - \frac{2}{t + 1}}{\frac{1}{4 - \frac{4}{{\left(t + 1\right)}^{2}}}}} \]

    expm1-log1p-u [=>]100.0%

    \[ 1 - \frac{1}{2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + \frac{2}{t + 1}} \cdot \frac{2 - \frac{2}{t + 1}}{\frac{1}{4 - \frac{4}{{\left(t + 1\right)}^{2}}}}\right)\right)}} \]

    expm1-udef [=>]100.0%

    \[ 1 - \frac{1}{2 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{2 + \frac{2}{t + 1}} \cdot \frac{2 - \frac{2}{t + 1}}{\frac{1}{4 - \frac{4}{{\left(t + 1\right)}^{2}}}}\right)} - 1\right)}} \]

  6. Simplified100.0%

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{2 + \frac{-2}{t + 1}}{\frac{2 + \frac{2}{t + 1}}{4 + -4 \cdot {\left(t + 1\right)}^{-2}}}}} \]
    Step-by-step derivation

    [Start]100.0%

    \[ 1 - \frac{1}{2 + \left(e^{\mathsf{log1p}\left(\frac{\left(2 - \frac{2}{t + 1}\right) \cdot \left(4 - 4 \cdot {\left(t + 1\right)}^{-2}\right)}{2 + \frac{2}{t + 1}}\right)} - 1\right)} \]

    expm1-def [=>]100.0%

    \[ 1 - \frac{1}{2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(2 - \frac{2}{t + 1}\right) \cdot \left(4 - 4 \cdot {\left(t + 1\right)}^{-2}\right)}{2 + \frac{2}{t + 1}}\right)\right)}} \]

    expm1-log1p [=>]100.0%

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

    associate-/l* [=>]100.0%

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

    /-rgt-identity [<=]100.0%

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

    /-rgt-identity [=>]100.0%

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

    sub-neg [=>]100.0%

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

    distribute-neg-frac [=>]100.0%

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

    metadata-eval [=>]100.0%

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

    cancel-sign-sub-inv [=>]100.0%

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

    metadata-eval [=>]100.0%

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

  7. Final simplification100.0%

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

Alternatives

Alternative 1
Accuracy100.0%
Cost8320
\[1 + \frac{-1}{2 + \frac{2 + \frac{-2}{1 + t}}{\frac{2 + \frac{2}{1 + t}}{4 + -4 \cdot {\left(1 + t\right)}^{-2}}}} \]
Alternative 2
Accuracy100.0%
Cost1856
\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 + \frac{-1}{2 + t_1 \cdot t_1} \end{array} \]
Alternative 3
Accuracy100.0%
Cost1216
\[1 + \frac{-1}{6 + \left(8 + \frac{-4}{1 + t}\right) \cdot \frac{1}{-1 - t}} \]
Alternative 4
Accuracy100.0%
Cost1088
\[1 + \frac{-1}{6 + \frac{\frac{4}{1 + t} + -8}{1 + t}} \]
Alternative 5
Accuracy96.8%
Cost836
\[\begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 6
Accuracy97.1%
Cost836
\[\begin{array}{l} \mathbf{if}\;t \leq -0.64:\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]
Alternative 7
Accuracy96.1%
Cost452
\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 8
Accuracy96.7%
Cost452
\[\begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 9
Accuracy96.0%
Cost196
\[\begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 10
Accuracy94.9%
Cost64
\[0.5 \]

Reproduce?

herbie shell --seed 2023277 
(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)))))))))