Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 10

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

\[\begin{array}{l} \\ 1 + \frac{-1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{1 + t}\right)}^{6}}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ 2.0 (cbrt (pow (+ 2.0 (/ -2.0 (+ 1.0 t))) 6.0))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (2.0 + cbrt(pow((2.0 + (-2.0 / (1.0 + t))), 6.0))));
}

Java

public static double code(double t) {
	return 1.0 + (-1.0 / (2.0 + Math.cbrt(Math.pow((2.0 + (-2.0 / (1.0 + t))), 6.0))));
}

Julia

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(2.0 + cbrt((Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))) ^ 6.0)))))
end

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(2.0 + N[Power[N[Power[N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right)}^{0.3333333333333333}}} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}}}} \]

   2. associate-/r* 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \color{blue}{\frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)}^{6}}} \]

   3. distribute-lft-in 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right)}^{6}}} \]

   4. *-rgt-identity 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t} + t \cdot \frac{1}{t}}\right)}^{6}}} \]

   5. rgt-mult-inverse 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{t + \color{blue}{1}}\right)}^{6}}} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{-2}{t + 1}\right)}^{6}}}} \]

7. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{1 + t}\right)}^{6}}} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \lor \neg \left(t \leq 0.8\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{-4}{1 + t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -1.1) (not (<= t 0.8)))
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))
   (+ 1.0 (/ -1.0 (+ 2.0 (* t (+ 4.0 (/ -4.0 (+ 1.0 t)))))))))

C

double code(double t) {
	double tmp;
	if ((t <= -1.1) || !(t <= 0.8)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.1d0)) .or. (.not. (t <= 0.8d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 - ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t)
    else
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (4.0d0 + ((-4.0d0) / (1.0d0 + t))))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -1.1) || !(t <= 0.8)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -1.1) or not (t <= 0.8):
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t)
	else:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -1.1) || !(t <= 0.8))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(4.0 + Float64(-4.0 / Float64(1.0 + t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -1.1) || ~((t <= 0.8)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	else
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -1.1], N[Not[LessEqual[t, 0.8]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(4.0 + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1000000000000001 or 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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   if -1.1000000000000001 < 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. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right)}^{0.3333333333333333}}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}}}} \]

      2. associate-/r* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \color{blue}{\frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)}^{6}}} \]

      3. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right)}^{6}}} \]

      4. *-rgt-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t} + t \cdot \frac{1}{t}}\right)}^{6}}} \]

      5. rgt-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{t + \color{blue}{1}}\right)}^{6}}} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{-2}{t + 1}\right)}^{6}}}} \]
   7. Step-by-step derivation

      1. pow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{-2}{t + 1}\right)}^{6}\right)}^{0.3333333333333333}}} \]

      2. pow-pow 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(2 + \frac{-2}{t + 1}\right)}^{\left(6 \cdot 0.3333333333333333\right)}}} \]

      3. metadata-eval 100.0%

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

      4. pow2 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in t around 0 99.2%

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

       1. *-commutative 99.2%

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

   11. Simplified 99.2%

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

       1. *-commutative 99.2%

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

       2. distribute-lft-in 99.2%

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

       3. +-commutative 99.2%

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

   13. Applied egg-rr 99.2%

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

       1. associate-*l* 99.2%

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

       2. metadata-eval 99.2%

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

       3. associate-*l* 99.2%

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

       4. distribute-lft-out 99.2%

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

       5. associate-*r/ 99.2%

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

       6. metadata-eval 99.2%

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

   15. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \lor \neg \left(t \leq 0.8\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{-4}{1 + t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{-4}{1 + t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.6)
   (+ 1.0 (/ -1.0 (- 6.0 (/ (- 8.0 (/ (+ 12.0 (/ -16.0 t)) t)) t))))
   (if (<= t 0.8)
     (+ 1.0 (/ -1.0 (+ 2.0 (* t (+ 4.0 (/ -4.0 (+ 1.0 t)))))))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 - ((12.0 + (-16.0 / t)) / t)) / t)));
	} else if (t <= 0.8) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))));
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / 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 + ((-1.0d0) / (6.0d0 - ((8.0d0 - ((12.0d0 + ((-16.0d0) / t)) / t)) / t)))
    else if (t <= 0.8d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (4.0d0 + ((-4.0d0) / (1.0d0 + t))))))
    else
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 - ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 - ((12.0 + (-16.0 / t)) / t)) / t)));
	} else if (t <= 0.8) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))));
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.6:
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 - ((12.0 + (-16.0 / t)) / t)) / t)))
	elif t <= 0.8:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))))
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 - Float64(Float64(8.0 - Float64(Float64(12.0 + Float64(-16.0 / t)) / t)) / t))));
	elseif (t <= 0.8)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(4.0 + Float64(-4.0 / Float64(1.0 + t)))))));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 - ((12.0 + (-16.0 / t)) / t)) / t)));
	elseif (t <= 0.8)
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (-4.0 / (1.0 + t))))));
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.6], N[(1.0 + N[(-1.0 / N[(6.0 - N[(N[(8.0 - N[(N[(12.0 + N[(-16.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.8], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(4.0 + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $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. Add Preprocessing

   3. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. associate-*r/ 98.7%

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

      7. metadata-eval 98.7%

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

      8. distribute-neg-frac 98.7%

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

      9. metadata-eval 98.7%

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

   5. Simplified 98.7%

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

   if -0.599999999999999978 < 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. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right)}^{0.3333333333333333}}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}}}} \]

      2. associate-/r* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \color{blue}{\frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)}^{6}}} \]

      3. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right)}^{6}}} \]

      4. *-rgt-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t} + t \cdot \frac{1}{t}}\right)}^{6}}} \]

      5. rgt-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{t + \color{blue}{1}}\right)}^{6}}} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{-2}{t + 1}\right)}^{6}}}} \]
   7. Step-by-step derivation

      1. pow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{-2}{t + 1}\right)}^{6}\right)}^{0.3333333333333333}}} \]

      2. pow-pow 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(2 + \frac{-2}{t + 1}\right)}^{\left(6 \cdot 0.3333333333333333\right)}}} \]

      3. metadata-eval 100.0%

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

      4. pow2 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in t around 0 99.7%

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

       1. *-commutative 99.7%

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

   11. Simplified 99.7%

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

       1. *-commutative 99.7%

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

       2. distribute-lft-in 99.7%

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

       3. +-commutative 99.7%

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

   13. Applied egg-rr 99.7%

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

       1. associate-*l* 99.7%

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

       2. metadata-eval 99.7%

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

       3. associate-*l* 99.7%

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

       4. distribute-lft-out 99.7%

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

       5. associate-*r/ 99.7%

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

       6. metadata-eval 99.7%

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

   15. Simplified 99.7%

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

   if 0.80000000000000004 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. unsub-neg 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{-4}{1 + t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.35 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.35) (not (<= t 0.7)))
   (+
    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 <= -0.35) || !(t <= 0.7)) {
		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 <= (-0.35d0)) .or. (.not. (t <= 0.7d0))) 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 <= -0.35) || !(t <= 0.7)) {
		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 <= -0.35) or not (t <= 0.7):
		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 <= -0.35) || !(t <= 0.7))
		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 <= -0.35) || ~((t <= 0.7)))
		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, -0.35], N[Not[LessEqual[t, 0.7]], $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 < -0.34999999999999998 or 0.69999999999999996 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

      5. associate-*r/ 99.1%

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

      6. metadata-eval 99.1%

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

   5. Simplified 99.1%

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

   if -0.34999999999999998 < t < 0.69999999999999996

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right)}^{0.3333333333333333}}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}}}} \]

      2. associate-/r* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \color{blue}{\frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)}^{6}}} \]

      3. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right)}^{6}}} \]

      4. *-rgt-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t} + t \cdot \frac{1}{t}}\right)}^{6}}} \]

      5. rgt-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{t + \color{blue}{1}}\right)}^{6}}} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{-2}{t + 1}\right)}^{6}}}} \]
   7. Step-by-step derivation

      1. pow1/3 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{-2}{t + 1}\right)}^{6}\right)}^{0.3333333333333333}}} \]

      2. pow-pow 100.0%

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(2 + \frac{-2}{t + 1}\right)}^{\left(6 \cdot 0.3333333333333333\right)}}} \]

      3. metadata-eval 100.0%

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

      4. pow2 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in t around 0 99.7%

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

       1. *-commutative 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in t around 0 99.7%

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

       1. *-commutative 99.7%

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

   14. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.33) (not (<= t 0.7)))
   (+
    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 <= 0.7)) {
		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 <= 0.7d0))) 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 <= 0.7)) {
		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 <= 0.7):
		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 <= 0.7))
		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 <= 0.7)))
		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, 0.7]], $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 0.69999999999999996 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

      5. associate-*r/ 99.1%

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

      6. metadata-eval 99.1%

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

   5. Simplified 99.1%

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

   if -0.330000000000000016 < t < 0.69999999999999996

   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 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.33 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right)}^{0.3333333333333333}}} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}}}} \]

   2. associate-/r* 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \color{blue}{\frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)}^{6}}} \]

   3. distribute-lft-in 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right)}^{6}}} \]

   4. *-rgt-identity 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{\color{blue}{t} + t \cdot \frac{1}{t}}\right)}^{6}}} \]

   5. rgt-mult-inverse 100.0%

      \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 + \frac{-2}{t + \color{blue}{1}}\right)}^{6}}} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{-2}{t + 1}\right)}^{6}}}} \]
7. Step-by-step derivation

   1. pow1/3 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{-2}{t + 1}\right)}^{6}\right)}^{0.3333333333333333}}} \]

   2. pow-pow 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(2 + \frac{-2}{t + 1}\right)}^{\left(6 \cdot 0.3333333333333333\right)}}} \]

   3. metadata-eval 100.0%

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

   4. pow2 100.0%

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

   5. +-commutative 100.0%

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

   6. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 7: 99.1% 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 98.9%

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

      1. sub-neg 98.9%

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

      2. associate-+l+ 98.9%

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

      3. sub-neg 98.9%

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

      4. unpow2 98.9%

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

      5. associate-/r* 98.9%

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

      6. metadata-eval 98.9%

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

      7. associate-*r/ 98.9%

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

      8. associate-*r/ 98.9%

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

      9. metadata-eval 98.9%

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

      10. div-sub 98.9%

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

      11. remove-double-neg 98.9%

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

      12. sub-neg 98.9%

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

      13. distribute-neg-in 98.9%

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

      14. metadata-eval 98.9%

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

      15. metadata-eval 98.9%

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

      16. +-commutative 98.9%

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

      17. sub-neg 98.9%

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

      18. distribute-neg-frac 98.9%

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

   5. Simplified 98.9%

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

   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 99.3%

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

4. Final simplification 99.1%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.65)) {
		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 <= 0.65d0))) 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 <= 0.65)) {
		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 <= 0.65):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.65))
		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 <= 0.65)))
		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, 0.65]], $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 0.650000000000000022 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   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 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% 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 97.4%

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

   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 99.3%

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

Alternative 10: 59.9% 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 61.7%

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

Reproduce

herbie shell --seed 2024167 
(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)))))))))