Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 14.2s

Alternatives: 9

Speedup: 1.3×

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, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{2 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(\left(3 - \frac{-2}{-1 - t}\right) + -1\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  1.0
  (/
   -1.0
   (+
    2.0
    (* (+ 2.0 (/ 2.0 (- -1.0 t))) (+ (- 3.0 (/ -2.0 (- -1.0 t))) -1.0))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * ((3.0 - (-2.0 / (-1.0 - t))) + -1.0))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 + (2.0d0 / ((-1.0d0) - t))) * ((3.0d0 - ((-2.0d0) / ((-1.0d0) - t))) + (-1.0d0)))))
end function

Java

public static double code(double t) {
	return 1.0 + (-1.0 / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * ((3.0 - (-2.0 / (-1.0 - t))) + -1.0))));
}

Python

def code(t):
	return 1.0 + (-1.0 / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * ((3.0 - (-2.0 / (-1.0 - t))) + -1.0))))

Julia

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 + Float64(2.0 / Float64(-1.0 - t))) * Float64(Float64(3.0 - Float64(-2.0 / Float64(-1.0 - t))) + -1.0)))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (-1.0 / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * ((3.0 - (-2.0 / (-1.0 - t))) + -1.0))));
end

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 - N[(-2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. div-inv 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
5. Step-by-step derivation

   1. unpow-1 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

   2. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

   3. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

   4. associate-*l* 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

   5. distribute-rgt-in 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

   6. *-lft-identity 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

   7. lft-mult-inverse 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 99.2%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right)\right)}} \]

   2. sub-neg 99.2%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{2 + \left(-\frac{1}{0.5 \cdot \left(t + 1\right)}\right)}\right)\right)} \]

   3. associate-/r* 99.2%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(2 + \left(-\color{blue}{\frac{\frac{1}{0.5}}{t + 1}}\right)\right)\right)} \]

   4. metadata-eval 99.2%

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

   5. distribute-neg-frac 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 + \frac{-2}{1 + t}\right)\right)}} \]
9. Step-by-step derivation

   1. expm1-undefine 99.2%

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

   2. sub-neg 99.2%

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

   3. log1p-undefine 100.0%

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

   4. rem-exp-log 100.0%

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

   5. associate-+r+ 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(\color{blue}{3} + \frac{-2}{1 + t}\right) + \left(-1\right)\right)} \]

   7. +-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(3 + \frac{-2}{\color{blue}{t + 1}}\right) + \left(-1\right)\right)} \]

   8. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + \color{blue}{-1}\right)} \]

10. Simplified 100.0%

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)}} \]
11. Step-by-step derivation

    1. clear-num 100.0%

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

    2. inv-pow 100.0%

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

    3. div-inv 100.0%

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

    4. clear-num 100.0%

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

    5. div-inv 100.0%

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

    6. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

12. Applied egg-rr 100.0%

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]
13. Step-by-step derivation

    1. unpow-1 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

    2. *-commutative 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

    3. *-commutative 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

    4. associate-*l* 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

    5. distribute-rgt-in 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

    6. *-lft-identity 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

    7. lft-mult-inverse 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

14. Simplified 100.0%

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]
15. Step-by-step derivation

    1. *-un-lft-identity 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{1 \cdot \frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

    2. associate-/r* 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - 1 \cdot \color{blue}{\frac{\frac{1}{0.5}}{t + 1}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

    3. metadata-eval 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - 1 \cdot \frac{\color{blue}{2}}{t + 1}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

    4. +-commutative 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - 1 \cdot \frac{2}{\color{blue}{1 + t}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

16. Applied egg-rr 100.0%

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{1 \cdot \frac{2}{1 + t}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]
17. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{2}{1 + t}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

    2. +-commutative 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{2}{\color{blue}{t + 1}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

18. Simplified 100.0%

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{2}{t + 1}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

19. Final simplification 100.0%

    \[\leadsto 1 + \frac{-1}{2 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(\left(3 - \frac{-2}{-1 - t}\right) + -1\right)} \]
20. Add Preprocessing

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 + \frac{1}{\left(2 \cdot t\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -1.15)
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (if (<= t 0.8)
     (+ 1.0 (/ 1.0 (- (* (* 2.0 t) (- (/ 2.0 (+ 1.0 t)) 2.0)) 2.0)))
     (+
      1.0
      (+
       -0.16666666666666666
       (/
        (-
         (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
         0.2222222222222222)
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -1.15) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.8) {
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.15d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.8d0) then
        tmp = 1.0d0 + (1.0d0 / (((2.0d0 * t) * ((2.0d0 / (1.0d0 + t)) - 2.0d0)) - 2.0d0))
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -1.15) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.8) {
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -1.15:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.8:
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0))
	else:
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -1.15)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.8)
		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)));
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -1.15)
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.8)
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	else
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -1.15], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.8], 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], N[(1.0 + N[(-0.16666666666666666 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1499999999999999

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1.1499999999999999 < t < 0.80000000000000004

   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. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

      4. associate-*l* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

      5. distribute-rgt-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

      6. *-lft-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

      7. lft-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 99.6%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right) \cdot \left(2 \cdot t\right)} \]
   10. Step-by-step derivation

       1. unpow-1 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

       2. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

       3. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

       4. associate-*l* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

       5. distribute-rgt-in 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

       6. *-lft-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

       7. lft-mult-inverse 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   11. Simplified 99.6%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(2 \cdot t\right)} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{1 \cdot \frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

       2. associate-/r* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - 1 \cdot \color{blue}{\frac{\frac{1}{0.5}}{t + 1}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - 1 \cdot \frac{\color{blue}{2}}{t + 1}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

       4. +-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - 1 \cdot \frac{2}{\color{blue}{1 + t}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

   13. Applied egg-rr 99.6%

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

       1. *-lft-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{2}{1 + t}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

       2. +-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{2}{\color{blue}{t + 1}}\right) \cdot \left(\left(3 + \frac{-2}{t + 1}\right) + -1\right)} \]

   15. Simplified 99.6%

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

   if 0.80000000000000004 < 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 98.8%

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

      1. sub-neg 98.8%

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

      2. +-commutative 98.8%

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

      3. mul-1-neg 98.8%

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

      4. unsub-neg 98.8%

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

      5. metadata-eval 98.8%

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

      6. mul-1-neg 98.8%

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

      7. unsub-neg 98.8%

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

      8. associate-*r/ 98.8%

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

      9. metadata-eval 98.8%

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

   7. Simplified 98.8%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 + \frac{1}{\left(2 \cdot t\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.6)
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (if (<= t 0.68)
     (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))
     (+
      1.0
      (+
       -0.16666666666666666
       (/
        (-
         (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
         0.2222222222222222)
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.68) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.6d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.68d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.68) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.6:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.68:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.68)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.68)
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 1.0 + (-0.16666666666666666 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.6], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.16666666666666666 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.599999999999999978 < 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. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

      4. associate-*l* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

      5. distribute-rgt-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

      6. *-lft-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

      7. lft-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 99.6%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right) \cdot \left(2 \cdot t\right)} \]
   10. Step-by-step derivation

       1. unpow-1 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

       2. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

       3. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

       4. associate-*l* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

       5. distribute-rgt-in 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

       6. *-lft-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

       7. lft-mult-inverse 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   11. Simplified 99.6%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(2 \cdot t\right)} \]

   12. Taylor expanded in t around 0 99.6%

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

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 98.8%

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

      1. sub-neg 98.8%

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

      2. +-commutative 98.8%

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

      3. mul-1-neg 98.8%

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

      4. unsub-neg 98.8%

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

      5. metadata-eval 98.8%

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

      6. mul-1-neg 98.8%

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

      7. unsub-neg 98.8%

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

      8. associate-*r/ 98.8%

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

      9. metadata-eval 98.8%

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

   7. Simplified 98.8%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.6)
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (if (<= t 0.44)
     (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))
     (+
      1.0
      (-
       -0.16666666666666666
       (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.44) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.6d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.44d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.44) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.6:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.44:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.44)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.44)
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.6], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.44], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.16666666666666666 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.599999999999999978 < t < 0.440000000000000002

   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. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

      4. associate-*l* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

      5. distribute-rgt-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

      6. *-lft-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

      7. lft-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 99.6%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right) \cdot \left(2 \cdot t\right)} \]
   10. Step-by-step derivation

       1. unpow-1 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

       2. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

       3. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

       4. associate-*l* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

       5. distribute-rgt-in 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(1 \cdot t + \frac{1}{t} \cdot t\right)}}\right)} \]

       6. *-lft-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + \frac{1}{t} \cdot t\right)}\right)} \]

       7. lft-mult-inverse 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   11. Simplified 99.6%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(2 \cdot t\right)} \]

   12. Taylor expanded in t around 0 99.6%

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

   if 0.440000000000000002 < 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 98.6%

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

      1. sub-neg 98.6%

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

      2. +-commutative 98.6%

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

      3. mul-1-neg 98.6%

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

      4. unsub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. sub-neg 98.6%

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

      7. associate-*r/ 98.6%

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

      8. metadata-eval 98.6%

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

      9. distribute-neg-frac 98.6%

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

      10. metadata-eval 98.6%

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

   7. Simplified 98.6%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (if (<= t 0.235)
     0.5
     (+
      1.0
      (-
       -0.16666666666666666
       (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.235) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.235d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.235) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.235:
		tmp = 0.5
	else:
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.235)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.235)
		tmp = 0.5;
	else
		tmp = 1.0 + (-0.16666666666666666 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.235], 0.5, N[(1.0 + N[(-0.16666666666666666 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.48999999999999999 < t < 0.23499999999999999

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.5%

      \[\leadsto 1 + \color{blue}{-0.5} \]

   if 0.23499999999999999 < 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 98.6%

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

      1. sub-neg 98.6%

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

      2. +-commutative 98.6%

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

      3. mul-1-neg 98.6%

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

      4. unsub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. sub-neg 98.6%

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

      7. associate-*r/ 98.6%

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

      8. metadata-eval 98.6%

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

      9. distribute-neg-frac 98.6%

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

      10. metadata-eval 98.6%

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

   7. Simplified 98.6%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \left(\frac{-0.2222222222222222}{t} + 0.8333333333333334\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (if (<= t 0.66)
     0.5
     (+ 1.0 (+ -1.0 (+ (/ -0.2222222222222222 t) 0.8333333333333334))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 + ((-0.2222222222222222 / t) + 0.8333333333333334));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) + (((-0.2222222222222222d0) / t) + 0.8333333333333334d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 + ((-0.2222222222222222 / t) + 0.8333333333333334));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 + ((-0.2222222222222222 / t) + 0.8333333333333334))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 + Float64(Float64(-0.2222222222222222 / t) + 0.8333333333333334)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 + ((-0.2222222222222222 / t) + 0.8333333333333334));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.66], 0.5, N[(1.0 + N[(-1.0 + N[(N[(-0.2222222222222222 / t), $MachinePrecision] + 0.8333333333333334), $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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.5%

      \[\leadsto 1 + \color{blue}{-0.5} \]

   if 0.660000000000000031 < 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\right)} \]

      2. expm1-undefine 98.3%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} - 1\right)} \]

      3. +-commutative 98.3%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.2222222222222222}{t} + -0.16666666666666666}\right)} - 1\right) \]

   9. Applied egg-rr 98.3%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. sub-neg 98.3%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} + \left(-1\right)\right)} \]

       2. metadata-eval 98.3%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)} + \color{blue}{-1}\right) \]

       3. +-commutative 98.3%

          \[\leadsto 1 + \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)}\right)} \]

       4. log1p-undefine 98.3%

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

       5. rem-exp-log 98.3%

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

       6. +-commutative 98.3%

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

       7. associate-+r+ 98.3%

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

       8. metadata-eval 98.3%

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

   11. Simplified 98.3%

       \[\leadsto 1 + \color{blue}{\left(-1 + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \left(\frac{-0.2222222222222222}{t} + 0.8333333333333334\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 1.0 + (-0.16666666666666666 + (-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 <= 0.66d0))) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-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 <= 0.66)) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.66):
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + 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 <= 0.66)))
		tmp = 1.0 + (-0.16666666666666666 + (-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, 0.66]], $MachinePrecision]], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. metadata-eval 99.1%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. associate-*r/ 99.1%

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

      6. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.5%

      \[\leadsto 1 + \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 98.3%

      \[\leadsto 1 + \color{blue}{-0.16666666666666666} \]

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.5%

      \[\leadsto 1 + \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

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

Alternative 9: 60.2% 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. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 57.1%

   \[\leadsto 1 + \color{blue}{-0.5} \]

6. Final simplification 57.1%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))))))))