Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 12

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{-2}{1 + t}\\ 1 + \frac{-1}{2 + {\left(\sqrt[3]{t\_1 \cdot t\_1}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ -2.0 (+ 1.0 t)))))
   (+ 1.0 (/ -1.0 (+ 2.0 (pow (cbrt (* t_1 t_1)) 3.0))))))

C

double code(double t) {
	double t_1 = 2.0 + (-2.0 / (1.0 + t));
	return 1.0 + (-1.0 / (2.0 + pow(cbrt((t_1 * t_1)), 3.0)));
}

Java

public static double code(double t) {
	double t_1 = 2.0 + (-2.0 / (1.0 + t));
	return 1.0 + (-1.0 / (2.0 + Math.pow(Math.cbrt((t_1 * t_1)), 3.0)));
}

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(-2.0 / Float64(1.0 + t)))
	return Float64(1.0 + Float64(-1.0 / Float64(2.0 + (cbrt(Float64(t_1 * t_1)) ^ 3.0))))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(2.0 + N[Power[N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

   1. add-cube-cbrt 100.0%

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

   2. pow3 100.0%

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

   3. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. distribute-lft-in 100.0%

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

    3. *-rgt-identity 100.0%

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

    4. rgt-mult-inverse 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.76 \lor \neg \left(t \leq 0.81\right):\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t \cdot \left(\frac{-4}{-1 - t} - 4\right) - 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.76) (not (<= t 0.81)))
   (-
    1.0
    (+
     (/
      (-
       0.2222222222222222
       (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
      t)
     0.16666666666666666))
   (+ 1.0 (/ 1.0 (- (* t (- (/ -4.0 (- -1.0 t)) 4.0)) 2.0)))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.76) || !(t <= 0.81)) {
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.76d0)) .or. (.not. (t <= 0.81d0))) then
        tmp = 1.0d0 - (((0.2222222222222222d0 - ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t) + 0.16666666666666666d0)
    else
        tmp = 1.0d0 + (1.0d0 / ((t * (((-4.0d0) / ((-1.0d0) - t)) - 4.0d0)) - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.76) || !(t <= 0.81)) {
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.76) or not (t <= 0.81):
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666)
	else:
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.76) || !(t <= 0.81))
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(t * Float64(Float64(-4.0 / Float64(-1.0 - t)) - 4.0)) - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.76) || ~((t <= 0.81)))
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	else
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.76], N[Not[LessEqual[t, 0.81]], $MachinePrecision]], N[(1.0 - N[(N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(N[(t * N[(N[(-4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.76000000000000001 or 0.81000000000000005 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. unsub-neg 98.5%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

      7. unpow2 98.5%

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

      8. associate-/r* 98.5%

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

      9. div-sub 98.5%

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

      10. associate-*r/ 98.5%

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

      11. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   if -0.76000000000000001 < t < 0.81000000000000005

   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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 99.1%

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

      1. *-commutative 99.1%

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

   9. Simplified 99.1%

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

       1. distribute-lft-in 99.1%

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

       2. *-commutative 99.1%

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

       3. *-commutative 99.1%

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

       4. +-commutative 99.1%

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

   11. Applied egg-rr 99.1%

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

       1. distribute-lft-in 99.1%

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

       2. *-commutative 99.1%

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

       3. associate-*r* 99.1%

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

       4. distribute-lft-in 99.1%

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

       5. metadata-eval 99.1%

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

       6. associate-*r/ 99.1%

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

       7. metadata-eval 99.1%

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

       8. +-commutative 99.1%

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

   13. Simplified 99.1%

       \[\leadsto 1 - \frac{1}{2 + \color{blue}{t \cdot \left(4 + \frac{-4}{t + 1}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.76 \lor \neg \left(t \leq 0.81\right):\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t \cdot \left(\frac{-4}{-1 - t} - 4\right) - 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.85\right):\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(t \cdot \left(4 + t \cdot -4\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.58) (not (<= t 0.85)))
   (-
    1.0
    (+
     (/
      (-
       0.2222222222222222
       (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
      t)
     0.16666666666666666))
   (+ 1.0 (/ -1.0 (+ 2.0 (* t (* t (+ 4.0 (* t -4.0)))))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.85)) {
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * (4.0 + (t * -4.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.58d0)) .or. (.not. (t <= 0.85d0))) then
        tmp = 1.0d0 - (((0.2222222222222222d0 - ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t) + 0.16666666666666666d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (t * (4.0d0 + (t * (-4.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.58) || !(t <= 0.85)) {
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * (4.0 + (t * -4.0))))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.58) or not (t <= 0.85):
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666)
	else:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * (4.0 + (t * -4.0))))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.58) || !(t <= 0.85))
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(t * Float64(4.0 + Float64(t * -4.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.58) || ~((t <= 0.85)))
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	else
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * (4.0 + (t * -4.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.58], N[Not[LessEqual[t, 0.85]], $MachinePrecision]], N[(1.0 - N[(N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(t * N[(4.0 + N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 0.849999999999999978 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. unsub-neg 98.5%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

      7. unpow2 98.5%

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

      8. associate-/r* 98.5%

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

      9. div-sub 98.5%

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

      10. associate-*r/ 98.5%

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

      11. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   if -0.57999999999999996 < t < 0.849999999999999978

   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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 99.1%

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

      1. *-commutative 99.1%

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

   9. Simplified 99.1%

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

       1. distribute-lft-in 99.1%

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

       2. *-commutative 99.1%

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

       3. *-commutative 99.1%

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

       4. +-commutative 99.1%

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

   11. Applied egg-rr 99.1%

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

       1. distribute-lft-in 99.1%

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

       2. *-commutative 99.1%

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

       3. associate-*r* 99.1%

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

       4. distribute-lft-in 99.1%

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

       5. metadata-eval 99.1%

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

       6. associate-*r/ 99.1%

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

       7. metadata-eval 99.1%

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

       8. +-commutative 99.1%

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

   13. Simplified 99.1%

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

   14. Taylor expanded in t around 0 99.1%

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

       1. *-commutative 99.1%

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

   16. Simplified 99.1%

       \[\leadsto 1 - \frac{1}{2 + t \cdot \color{blue}{\left(t \cdot \left(4 + t \cdot -4\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58 \lor \neg \left(t \leq 0.85\right):\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(t \cdot \left(4 + t \cdot -4\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.68)))
   (-
    1.0
    (+
     (/
      (-
       0.2222222222222222
       (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
      t)
     0.16666666666666666))
   (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.68)) {
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 1.0d0 - (((0.2222222222222222d0 - ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t) + 0.16666666666666666d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.68)) {
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.68):
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666)
	else:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.68))
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.68)))
		tmp = 1.0 - (((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t) + 0.16666666666666666);
	else
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(1.0 - N[(N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. unsub-neg 98.5%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

      7. unpow2 98.5%

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

      8. associate-/r* 98.5%

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

      9. div-sub 98.5%

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

      10. associate-*r/ 98.5%

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

      11. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   if -0.52000000000000002 < 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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 99.1%

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

      1. *-commutative 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in t around 0 99.1%

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

       1. *-commutative 99.1%

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

   12. Simplified 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.68)))
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.68):
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

      5. associate-*r/ 98.4%

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

      6. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.52000000000000002 < 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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 99.1%

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

      1. *-commutative 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in t around 0 99.1%

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

       1. *-commutative 99.1%

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

   12. Simplified 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.33) (not (<= t 0.68)))
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.33) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 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 <= (-0.33d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.33) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.33) or not (t <= 0.68):
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.33) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.33) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.33], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

      5. associate-*r/ 98.4%

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

      6. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.330000000000000016 < 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. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.33 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.9%

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

      1. associate--l+ 97.9%

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

      2. unpow2 97.9%

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

      3. associate-/r* 97.9%

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

      4. metadata-eval 97.9%

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

      5. associate-*r/ 97.9%

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

      6. associate-*r/ 97.9%

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

      7. metadata-eval 97.9%

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

      8. div-sub 97.9%

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

      9. sub-neg 97.9%

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

      10. metadata-eval 97.9%

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

      11. metadata-eval 97.9%

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

      12. distribute-neg-in 97.9%

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

      13. +-commutative 97.9%

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

      14. sub-neg 97.9%

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

      15. distribute-neg-frac 97.9%

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

      16. unsub-neg 97.9%

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

   5. Simplified 97.9%

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

   if -0.52000000000000002 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.4%

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

Alternative 8: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

Python

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

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(-2.0 / 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 / (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[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $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. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 9: 99.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -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 <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.9%

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

      1. sub-neg 97.9%

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

      2. associate-+l+ 97.9%

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

      3. sub-neg 97.9%

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

      4. unpow2 97.9%

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

      5. associate-/r* 97.9%

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

      6. metadata-eval 97.9%

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

      7. associate-*r/ 97.9%

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

      8. associate-*r/ 97.9%

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

      9. metadata-eval 97.9%

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

      10. div-sub 97.9%

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

      11. remove-double-neg 97.9%

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

      12. sub-neg 97.9%

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

      13. distribute-neg-in 97.9%

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

      14. metadata-eval 97.9%

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

      15. metadata-eval 97.9%

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

      16. +-commutative 97.9%

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

      17. sub-neg 97.9%

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

      18. distribute-neg-frac 97.9%

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

   5. Simplified 97.9%

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

   if -0.52000000000000002 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.4%

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

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

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

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < 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. Add Preprocessing

   3. Taylor expanded in t around inf 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   5. Simplified 97.4%

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

   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. Add Preprocessing

   3. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.1%

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

Alternative 11: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		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 <= (-0.33d0)) 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 <= -0.33) {
		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 <= -0.33:
		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 <= -0.33)
		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 <= -0.33)
		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, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 96.1%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.8%

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

Alternative 12: 59.6% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0 59.2%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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)))))))))