Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 14.9s

Alternatives: 12

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (+
  1.0
  (/
   1.0
   (-
    (*
     (+ 2.0 (/ -2.0 (+ 1.0 t)))
     (- (/ (+ 2.0 (/ -2.0 t)) (+ t (/ -1.0 t))) 2.0))
    2.0))))

C

double code(double t) {
	return 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * (((2.0 + (-2.0 / t)) / (t + (-1.0 / t))) - 2.0)) - 2.0));
}

Fortran

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

Java

public static double code(double t) {
	return 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * (((2.0 + (-2.0 / t)) / (t + (-1.0 / t))) - 2.0)) - 2.0));
}

Python

def code(t):
	return 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * (((2.0 + (-2.0 / t)) / (t + (-1.0 / t))) - 2.0)) - 2.0))

Julia

function code(t)
	return Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))) * Float64(Float64(Float64(2.0 + Float64(-2.0 / t)) / Float64(t + Float64(-1.0 / t))) - 2.0)) - 2.0)))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * (((2.0 + (-2.0 / t)) / (t + (-1.0 / t))) - 2.0)) - 2.0));
end

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 + N[(-2.0 / t), $MachinePrecision]), $MachinePrecision] / N[(t + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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. Add Preprocessing
3. Step-by-step derivation

   1. flip-+ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. inv-pow 100.0%

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

   5. inv-pow 100.0%

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

   6. pow-prod-up 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/l/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-rgt1-in 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-*r/ 100.0%

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

   10. metadata-eval 100.0%

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

   11. distribute-neg-frac 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in t around 0 100.0%

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

   1. div-sub 100.0%

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

   2. sub-neg 100.0%

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

   3. unpow2 100.0%

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

   4. associate-/l* 100.0%

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

   5. *-inverses 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. metadata-eval 100.0%

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

9. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. metadata-eval 100.0%

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

11. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. distribute-rgt-in 100.0%

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

    3. lft-mult-inverse 100.0%

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

    4. *-lft-identity 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

    \[\leadsto 1 + \frac{1}{\left(2 + \frac{-2}{1 + t}\right) \cdot \left(\frac{2 + \frac{-2}{t}}{t + \frac{-1}{t}} - 2\right) - 2} \]
15. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 - \frac{-1}{t}} \leq 1.5:\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\left(2 \cdot t\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- 1.0 (/ -1.0 t))) 1.5)
   (+
    1.0
    (-
     (/
      (-
       (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
       0.2222222222222222)
      t)
     0.16666666666666666))
   (+ 1.0 (/ 1.0 (- (* (* 2.0 t) (- (/ 2.0 (+ 1.0 t)) 2.0)) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 - (-1.0 / t))) <= 1.5) {
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 - ((-1.0d0) / t))) <= 1.5d0) then
        tmp = 1.0d0 + (((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t) - 0.16666666666666666d0)
    else
        tmp = 1.0d0 + (1.0d0 / (((2.0d0 * t) * ((2.0d0 / (1.0d0 + t)) - 2.0d0)) - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 - (-1.0 / t))) <= 1.5) {
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 - (-1.0 / t))) <= 1.5:
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666)
	else:
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 - Float64(-1.0 / t))) <= 1.5)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(2.0 * t) * Float64(Float64(2.0 / Float64(1.0 + t)) - 2.0)) - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 - (-1.0 / t))) <= 1.5)
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	else
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 - N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], N[(1.0 + N[(N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(N[(N[(2.0 * t), $MachinePrecision] * N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1.5

   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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   5. Simplified 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

      2. associate-+r- 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{\left(2 + 4\right) - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

      3. metadata-eval 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{6} - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}} \]

   7. Applied egg-rr 99.3%

      \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   9. Simplified 99.3%

      \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   10. Taylor expanded in t around inf 99.4%

       \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(-1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}} + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(-1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}} + \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

       2. metadata-eval 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(-1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}} + \frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

       3. +-commutative 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(\frac{0.2222222222222222}{t} + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}}\right)}\right) \]

       4. mul-1-neg 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}}\right)}\right)\right) \]

       5. unsub-neg 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}}\right)}\right) \]

       6. unpow2 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{\color{blue}{t \cdot t}}\right)\right) \]

       7. associate-/r* 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}}\right)\right) \]

       8. div-sub 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}}\right) \]

       9. associate-*r/ 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t}\right) \]

       10. metadata-eval 99.4%

           \[\leadsto 1 - \left(0.16666666666666666 + \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t}\right) \]

   12. Simplified 99.4%

       \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\right)} \]

   if 1.5 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 99.1%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(2 \cdot t\right)} \cdot \left(2 - \frac{2}{t + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 - \frac{-1}{t}} \leq 1.5:\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\left(2 \cdot t\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.34) (not (<= t 0.68)))
   (+
    1.0
    (-
     (/
      (-
       (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
       0.2222222222222222)
      t)
     0.16666666666666666))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.34) || !(t <= 0.68)) {
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.34d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 1.0d0 + (((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t) - 0.16666666666666666d0)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.34) || !(t <= 0.68)) {
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.34) or not (t <= 0.68):
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.34) || !(t <= 0.68))
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.34) || ~((t <= 0.68)))
		tmp = 1.0 + (((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t) - 0.16666666666666666);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.34], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(1.0 + N[(N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 0.680000000000000049 < t

   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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   5. Simplified 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

      2. associate-+r- 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{\left(2 + 4\right) - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

      3. metadata-eval 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{6} - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}} \]

   7. Applied egg-rr 99.3%

      \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   9. Simplified 99.3%

      \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   10. Taylor expanded in t around inf 99.4%

       \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(-1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}} + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(-1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}} + \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

       2. metadata-eval 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(-1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}} + \frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

       3. +-commutative 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(\frac{0.2222222222222222}{t} + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}}\right)}\right) \]

       4. mul-1-neg 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}}\right)}\right)\right) \]

       5. unsub-neg 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{{t}^{2}}\right)}\right) \]

       6. unpow2 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{\color{blue}{t \cdot t}}\right)\right) \]

       7. associate-/r* 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \left(\frac{0.2222222222222222}{t} - \color{blue}{\frac{\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}}\right)\right) \]

       8. div-sub 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}}\right) \]

       9. associate-*r/ 99.4%

          \[\leadsto 1 - \left(0.16666666666666666 + \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t}\right) \]

       10. metadata-eval 99.4%

           \[\leadsto 1 - \left(0.16666666666666666 + \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t}\right) \]

   12. Simplified 99.4%

       \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\right)} \]

   if -0.340000000000000024 < t < 0.680000000000000049

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 97.3%

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

   8. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.34) (not (<= t 0.68)))
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.34) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.34d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 - ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.34) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.34) or not (t <= 0.68):
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.34) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.34) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.34], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 0.680000000000000049 < t

   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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   5. Simplified 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

      2. associate-+r- 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{\left(2 + 4\right) - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

      3. metadata-eval 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{6} - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}} \]

   7. Applied egg-rr 99.3%

      \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   9. Simplified 99.3%

      \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   10. Taylor expanded in t around -inf 99.4%

       \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   11. Step-by-step derivation

       1. mul-1-neg 99.4%

          \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

       2. unsub-neg 99.4%

          \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

       3. mul-1-neg 99.4%

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

       4. unsub-neg 99.4%

          \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

       5. associate-*r/ 99.4%

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

       6. metadata-eval 99.4%

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   12. Simplified 99.4%

       \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]

   if -0.340000000000000024 < t < 0.680000000000000049

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 97.3%

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

   8. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.23)))
   (+
    1.0
    (-
     (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)
     0.16666666666666666))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 1.0d0 + ((((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t) - 0.16666666666666666d0)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.23):
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.23))
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.23)))
		tmp = 1.0 + ((((0.037037037037037035 / t) + -0.2222222222222222) / t) - 0.16666666666666666);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(1.0 + N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + \frac{12}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(\frac{12}{{t}^{2}} + 4\right)} - 8 \cdot \frac{1}{t}\right)} \]

      2. associate--l+ 99.1%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\frac{12}{{t}^{2}} + \left(4 - 8 \cdot \frac{1}{t}\right)\right)}} \]

      3. +-commutative 99.1%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 - 8 \cdot \frac{1}{t}\right) + \frac{12}{{t}^{2}}\right)}} \]

      4. associate--r- 99.1%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)\right)}} \]

      5. associate-*r/ 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)\right)} \]

      7. unpow2 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{12}{\color{blue}{t \cdot t}}\right)\right)} \]

      8. associate-/r* 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \color{blue}{\frac{\frac{12}{t}}{t}}\right)\right)} \]

      9. metadata-eval 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t}\right)\right)} \]

      10. associate-*r/ 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\color{blue}{12 \cdot \frac{1}{t}}}{t}\right)\right)} \]

      11. div-sub 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \color{blue}{\frac{8 - 12 \cdot \frac{1}{t}}{t}}\right)} \]

      12. sub-neg 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)} \]

      13. associate-*r/ 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)} \]

      14. metadata-eval 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)} \]

      15. distribute-neg-frac 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)} \]

      16. metadata-eval 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)} \]

   5. Simplified 99.1%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

   6. Taylor expanded in t around -inf 99.2%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + -1 \cdot \frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{t}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.2%

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\left(-\frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{t}\right)}\right) \]

      2. unsub-neg 99.2%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 - \frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{t}\right)} \]

      3. sub-neg 99.2%

         \[\leadsto 1 - \left(0.16666666666666666 - \frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t} + \left(-0.2222222222222222\right)}}{t}\right) \]

      4. associate-*r/ 99.2%

         \[\leadsto 1 - \left(0.16666666666666666 - \frac{\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}} + \left(-0.2222222222222222\right)}{t}\right) \]

      5. metadata-eval 99.2%

         \[\leadsto 1 - \left(0.16666666666666666 - \frac{\frac{\color{blue}{0.037037037037037035}}{t} + \left(-0.2222222222222222\right)}{t}\right) \]

      6. metadata-eval 99.2%

         \[\leadsto 1 - \left(0.16666666666666666 - \frac{\frac{0.037037037037037035}{t} + \color{blue}{-0.2222222222222222}}{t}\right) \]

   8. Simplified 99.2%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 - \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)} \]

   if -0.52000000000000002 < t < 0.23000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 97.3%

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

   8. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;1 + \left(\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} - 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ 1 + \frac{1}{\left(2 - t\_1\right) \cdot \left(t\_1 - 2\right) - 2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (+ 1.0 (/ 1.0 (- (* (- 2.0 t_1) (- t_1 2.0)) 2.0)))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (1.0 / (((2.0 - t_1) * (t_1 - 2.0)) - 2.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 / (1.0d0 + t)
    code = 1.0d0 + (1.0d0 / (((2.0d0 - t_1) * (t_1 - 2.0d0)) - 2.0d0))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (1.0 / (((2.0 - t_1) * (t_1 - 2.0)) - 2.0));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	return 1.0 + (1.0 / (((2.0 - t_1) * (t_1 - 2.0)) - 2.0))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	return Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(2.0 - t_1) * Float64(t_1 - 2.0)) - 2.0)))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	tmp = 1.0 + (1.0 / (((2.0 - t_1) * (t_1 - 2.0)) - 2.0));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(1.0 / N[(N[(N[(2.0 - t$95$1), $MachinePrecision] * N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. associate-/l/ 100.0%

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

   7. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. lft-mult-inverse 100.0%

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

   4. *-lft-identity 100.0%

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

6. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. associate-/l/ 100.0%

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

   7. *-commutative 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
9. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. lft-mult-inverse 100.0%

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

   4. *-lft-identity 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

    \[\leadsto 1 + \frac{1}{\left(2 - \frac{2}{1 + t}\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2} \]
12. Add Preprocessing

Alternative 7: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.23)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.23):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.23))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.23)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 99.3%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   5. Simplified 99.3%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{2 + \left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

      2. associate-+r- 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{\left(2 + 4\right) - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

      3. metadata-eval 99.3%

         \[\leadsto 1 - 1 \cdot \frac{1}{\color{blue}{6} - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}} \]

   7. Applied egg-rr 99.3%

      \[\leadsto 1 - \color{blue}{1 \cdot \frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 99.3%

         \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   9. Simplified 99.3%

      \[\leadsto 1 - \color{blue}{\frac{1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   10. Taylor expanded in t around inf 99.2%

       \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.2%

          \[\leadsto \left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

       2. metadata-eval 99.2%

          \[\leadsto \left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - \frac{\color{blue}{0.2222222222222222}}{t} \]

       3. associate-+r- 99.2%

          \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{0.037037037037037035}{{t}^{2}} - \frac{0.2222222222222222}{t}\right)} \]

       4. unpow2 99.2%

          \[\leadsto 0.8333333333333334 + \left(\frac{0.037037037037037035}{\color{blue}{t \cdot t}} - \frac{0.2222222222222222}{t}\right) \]

       5. associate-/r* 99.2%

          \[\leadsto 0.8333333333333334 + \left(\color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}} - \frac{0.2222222222222222}{t}\right) \]

       6. metadata-eval 99.2%

          \[\leadsto 0.8333333333333334 + \left(\frac{\frac{\color{blue}{0.037037037037037035 \cdot 1}}{t}}{t} - \frac{0.2222222222222222}{t}\right) \]

       7. associate-*r/ 99.2%

          \[\leadsto 0.8333333333333334 + \left(\frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t}}}{t} - \frac{0.2222222222222222}{t}\right) \]

       8. div-sub 99.2%

          \[\leadsto 0.8333333333333334 + \color{blue}{\frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{t}} \]

       9. remove-double-neg 99.2%

          \[\leadsto 0.8333333333333334 + \frac{\color{blue}{-\left(-\left(0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222\right)\right)}}{t} \]

       10. sub-neg 99.2%

           \[\leadsto 0.8333333333333334 + \frac{-\left(-\color{blue}{\left(0.037037037037037035 \cdot \frac{1}{t} + \left(-0.2222222222222222\right)\right)}\right)}{t} \]

       11. metadata-eval 99.2%

           \[\leadsto 0.8333333333333334 + \frac{-\left(-\left(0.037037037037037035 \cdot \frac{1}{t} + \color{blue}{-0.2222222222222222}\right)\right)}{t} \]

       12. +-commutative 99.2%

           \[\leadsto 0.8333333333333334 + \frac{-\left(-\color{blue}{\left(-0.2222222222222222 + 0.037037037037037035 \cdot \frac{1}{t}\right)}\right)}{t} \]

       13. distribute-neg-in 99.2%

           \[\leadsto 0.8333333333333334 + \frac{-\color{blue}{\left(\left(--0.2222222222222222\right) + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)\right)}}{t} \]

       14. metadata-eval 99.2%

           \[\leadsto 0.8333333333333334 + \frac{-\left(\color{blue}{0.2222222222222222} + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)\right)}{t} \]

       15. sub-neg 99.2%

           \[\leadsto 0.8333333333333334 + \frac{-\color{blue}{\left(0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} \]

       16. distribute-neg-frac 99.2%

           \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}\right)} \]

       17. unsub-neg 99.2%

           \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]

   12. Simplified 99.2%

       \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}} \]

   if -0.52000000000000002 < t < 0.23000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 97.3%

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

   8. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (if (<= t 0.68)
     0.5
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.7%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + \frac{12}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(\frac{12}{{t}^{2}} + 4\right)} - 8 \cdot \frac{1}{t}\right)} \]

      2. associate--l+ 99.7%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\frac{12}{{t}^{2}} + \left(4 - 8 \cdot \frac{1}{t}\right)\right)}} \]

      3. +-commutative 99.7%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 - 8 \cdot \frac{1}{t}\right) + \frac{12}{{t}^{2}}\right)}} \]

      4. associate--r- 99.7%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)\right)}} \]

      5. associate-*r/ 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)\right)} \]

      7. unpow2 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{12}{\color{blue}{t \cdot t}}\right)\right)} \]

      8. associate-/r* 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \color{blue}{\frac{\frac{12}{t}}{t}}\right)\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t}\right)\right)} \]

      10. associate-*r/ 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\color{blue}{12 \cdot \frac{1}{t}}}{t}\right)\right)} \]

      11. div-sub 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \color{blue}{\frac{8 - 12 \cdot \frac{1}{t}}{t}}\right)} \]

      12. sub-neg 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)} \]

      13. associate-*r/ 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)} \]

      14. metadata-eval 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)} \]

      15. distribute-neg-frac 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)} \]

      16. metadata-eval 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)} \]

   5. Simplified 99.7%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

   6. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

   7. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{\frac{0.8333333333333334 \cdot t - 0.2222222222222222}{t}} \]
   8. Step-by-step derivation

      1. div-sub 98.9%

         \[\leadsto \color{blue}{\frac{0.8333333333333334 \cdot t}{t} - \frac{0.2222222222222222}{t}} \]

      2. sub-neg 98.9%

         \[\leadsto \color{blue}{\frac{0.8333333333333334 \cdot t}{t} + \left(-\frac{0.2222222222222222}{t}\right)} \]

      3. associate-/l* 99.1%

         \[\leadsto \color{blue}{0.8333333333333334 \cdot \frac{t}{t}} + \left(-\frac{0.2222222222222222}{t}\right) \]

      4. *-inverses 99.1%

         \[\leadsto 0.8333333333333334 \cdot \color{blue}{1} + \left(-\frac{0.2222222222222222}{t}\right) \]

      5. metadata-eval 99.1%

         \[\leadsto \color{blue}{0.8333333333333334} + \left(-\frac{0.2222222222222222}{t}\right) \]

      6. distribute-neg-frac 99.1%

         \[\leadsto 0.8333333333333334 + \color{blue}{\frac{-0.2222222222222222}{t}} \]

      7. metadata-eval 99.1%

         \[\leadsto 0.8333333333333334 + \frac{\color{blue}{-0.2222222222222222}}{t} \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 97.3%

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

   8. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{0.5} \]

   if 0.680000000000000049 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf 98.6%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + \frac{12}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(\frac{12}{{t}^{2}} + 4\right)} - 8 \cdot \frac{1}{t}\right)} \]

      2. associate--l+ 98.6%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\frac{12}{{t}^{2}} + \left(4 - 8 \cdot \frac{1}{t}\right)\right)}} \]

      3. +-commutative 98.6%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 - 8 \cdot \frac{1}{t}\right) + \frac{12}{{t}^{2}}\right)}} \]

      4. associate--r- 98.6%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)\right)}} \]

      5. associate-*r/ 98.6%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)\right)} \]

      7. unpow2 98.6%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{12}{\color{blue}{t \cdot t}}\right)\right)} \]

      8. associate-/r* 98.6%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \color{blue}{\frac{\frac{12}{t}}{t}}\right)\right)} \]

      9. metadata-eval 98.6%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t}\right)\right)} \]

      10. associate-*r/ 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\color{blue}{12 \cdot \frac{1}{t}}}{t}\right)\right)} \]

      11. div-sub 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \color{blue}{\frac{8 - 12 \cdot \frac{1}{t}}{t}}\right)} \]

      12. sub-neg 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)} \]

      13. associate-*r/ 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)} \]

      14. metadata-eval 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)} \]

      15. distribute-neg-frac 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)} \]

      16. metadata-eval 98.6%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)} \]

   5. Simplified 98.6%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

   6. Taylor expanded in t around inf 98.5%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.5%

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) \]

      2. metadata-eval 98.5%

         \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.2222222222222222}}{t}\right) \]

   8. Simplified 98.5%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.68):
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + \frac{12}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(\frac{12}{{t}^{2}} + 4\right)} - 8 \cdot \frac{1}{t}\right)} \]

      2. associate--l+ 99.1%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\frac{12}{{t}^{2}} + \left(4 - 8 \cdot \frac{1}{t}\right)\right)}} \]

      3. +-commutative 99.1%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 - 8 \cdot \frac{1}{t}\right) + \frac{12}{{t}^{2}}\right)}} \]

      4. associate--r- 99.1%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)\right)}} \]

      5. associate-*r/ 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)\right)} \]

      7. unpow2 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{12}{\color{blue}{t \cdot t}}\right)\right)} \]

      8. associate-/r* 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \color{blue}{\frac{\frac{12}{t}}{t}}\right)\right)} \]

      9. metadata-eval 99.1%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t}\right)\right)} \]

      10. associate-*r/ 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\color{blue}{12 \cdot \frac{1}{t}}}{t}\right)\right)} \]

      11. div-sub 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \color{blue}{\frac{8 - 12 \cdot \frac{1}{t}}{t}}\right)} \]

      12. sub-neg 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)} \]

      13. associate-*r/ 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)} \]

      14. metadata-eval 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)} \]

      15. distribute-neg-frac 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)} \]

      16. metadata-eval 99.1%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)} \]

   5. Simplified 99.1%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

   6. Taylor expanded in t around inf 98.8%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

   7. Taylor expanded in t around 0 98.5%

      \[\leadsto \color{blue}{\frac{0.8333333333333334 \cdot t - 0.2222222222222222}{t}} \]
   8. Step-by-step derivation

      1. div-sub 98.5%

         \[\leadsto \color{blue}{\frac{0.8333333333333334 \cdot t}{t} - \frac{0.2222222222222222}{t}} \]

      2. sub-neg 98.5%

         \[\leadsto \color{blue}{\frac{0.8333333333333334 \cdot t}{t} + \left(-\frac{0.2222222222222222}{t}\right)} \]

      3. associate-/l* 98.8%

         \[\leadsto \color{blue}{0.8333333333333334 \cdot \frac{t}{t}} + \left(-\frac{0.2222222222222222}{t}\right) \]

      4. *-inverses 98.8%

         \[\leadsto 0.8333333333333334 \cdot \color{blue}{1} + \left(-\frac{0.2222222222222222}{t}\right) \]

      5. metadata-eval 98.8%

         \[\leadsto \color{blue}{0.8333333333333334} + \left(-\frac{0.2222222222222222}{t}\right) \]

      6. distribute-neg-frac 98.8%

         \[\leadsto 0.8333333333333334 + \color{blue}{\frac{-0.2222222222222222}{t}} \]

      7. metadata-eval 98.8%

         \[\leadsto 0.8333333333333334 + \frac{\color{blue}{-0.2222222222222222}}{t} \]

   9. Simplified 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 97.3%

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

   8. Taylor expanded in t around 0 98.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ 2.0 (* 2.0 (- 2.0 (/ 2.0 (+ 1.0 t))))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (2.0 + (2.0 * (2.0 - (2.0 / (1.0 + t))))));
}

Fortran

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

Java

public static double code(double t) {
	return 1.0 + (-1.0 / (2.0 + (2.0 * (2.0 - (2.0 / (1.0 + t))))));
}

Python

def code(t):
	return 1.0 + (-1.0 / (2.0 + (2.0 * (2.0 - (2.0 / (1.0 + t))))))

Julia

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(2.0 * Float64(2.0 - Float64(2.0 / Float64(1.0 + t)))))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (-1.0 / (2.0 + (2.0 * (2.0 - (2.0 / (1.0 + t))))));
end

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(2.0 + N[(2.0 * N[(2.0 - N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. associate-/l/ 100.0%

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

   7. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. lft-mult-inverse 100.0%

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

   4. *-lft-identity 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in t around inf 97.5%

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

8. Final simplification 97.5%

   \[\leadsto 1 + \frac{-1}{2 + 2 \cdot \left(2 - \frac{2}{1 + t}\right)} \]
9. Add Preprocessing

Alternative 11: 98.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.7%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + \frac{12}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(\frac{12}{{t}^{2}} + 4\right)} - 8 \cdot \frac{1}{t}\right)} \]

      2. associate--l+ 99.7%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\frac{12}{{t}^{2}} + \left(4 - 8 \cdot \frac{1}{t}\right)\right)}} \]

      3. +-commutative 99.7%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 - 8 \cdot \frac{1}{t}\right) + \frac{12}{{t}^{2}}\right)}} \]

      4. associate--r- 99.7%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)\right)}} \]

      5. associate-*r/ 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)\right)} \]

      7. unpow2 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{12}{\color{blue}{t \cdot t}}\right)\right)} \]

      8. associate-/r* 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \color{blue}{\frac{\frac{12}{t}}{t}}\right)\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t}\right)\right)} \]

      10. associate-*r/ 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \left(\frac{8}{t} - \frac{\color{blue}{12 \cdot \frac{1}{t}}}{t}\right)\right)} \]

      11. div-sub 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \color{blue}{\frac{8 - 12 \cdot \frac{1}{t}}{t}}\right)} \]

      12. sub-neg 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)} \]

      13. associate-*r/ 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)} \]

      14. metadata-eval 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)} \]

      15. distribute-neg-frac 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)} \]

      16. metadata-eval 99.7%

          \[\leadsto 1 - \frac{1}{2 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)} \]

   5. Simplified 99.7%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

   6. Taylor expanded in t around inf 99.3%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

   7. Taylor expanded in t around inf 98.2%

      \[\leadsto \color{blue}{0.8333333333333334} \]

   if -0.340000000000000024 < t < 1

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. associate-/l/ 100.0%

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

      7. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. lft-mult-inverse 100.0%

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

      4. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 96.7%

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

   8. Taylor expanded in t around 0 98.2%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. associate-/l/ 100.0%

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

   7. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. lft-mult-inverse 100.0%

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

   4. *-lft-identity 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in t around inf 97.5%

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

8. Taylor expanded in t around 0 56.8%

   \[\leadsto \color{blue}{0.5} \]

9. Final simplification 56.8%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))))))))