Kahan p13 Example 3

Percentage Accurate: 99.9% → 100.0%

Time: 16.3s

Alternatives: 9

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: 99.9% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{\frac{8 + \frac{-8}{{\left(1 + t\right)}^{3}}}{4 + \frac{4 + \frac{4}{1 + t}}{1 + t}} \cdot \left(\frac{-2}{-1 - t} - 2\right) - 2} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  1.0
  (/
   1.0
   (-
    (*
     (/
      (+ 8.0 (/ -8.0 (pow (+ 1.0 t) 3.0)))
      (+ 4.0 (/ (+ 4.0 (/ 4.0 (+ 1.0 t))) (+ 1.0 t))))
     (- (/ -2.0 (- -1.0 t)) 2.0))
    2.0))))

C

double code(double t) {
	return 1.0 + (1.0 / ((((8.0 + (-8.0 / pow((1.0 + t), 3.0))) / (4.0 + ((4.0 + (4.0 / (1.0 + t))) / (1.0 + t)))) * ((-2.0 / (-1.0 - t)) - 2.0)) - 2.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + (1.0d0 / ((((8.0d0 + ((-8.0d0) / ((1.0d0 + t) ** 3.0d0))) / (4.0d0 + ((4.0d0 + (4.0d0 / (1.0d0 + t))) / (1.0d0 + t)))) * (((-2.0d0) / ((-1.0d0) - t)) - 2.0d0)) - 2.0d0))
end function

Java

public static double code(double t) {
	return 1.0 + (1.0 / ((((8.0 + (-8.0 / Math.pow((1.0 + t), 3.0))) / (4.0 + ((4.0 + (4.0 / (1.0 + t))) / (1.0 + t)))) * ((-2.0 / (-1.0 - t)) - 2.0)) - 2.0));
}

Python

def code(t):
	return 1.0 + (1.0 / ((((8.0 + (-8.0 / math.pow((1.0 + t), 3.0))) / (4.0 + ((4.0 + (4.0 / (1.0 + t))) / (1.0 + t)))) * ((-2.0 / (-1.0 - t)) - 2.0)) - 2.0))

Julia

function code(t)
	return Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(Float64(8.0 + Float64(-8.0 / (Float64(1.0 + t) ^ 3.0))) / Float64(4.0 + Float64(Float64(4.0 + Float64(4.0 / Float64(1.0 + t))) / Float64(1.0 + t)))) * Float64(Float64(-2.0 / Float64(-1.0 - t)) - 2.0)) - 2.0)))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (1.0 / ((((8.0 + (-8.0 / ((1.0 + t) ^ 3.0))) / (4.0 + ((4.0 + (4.0 / (1.0 + t))) / (1.0 + t)))) * ((-2.0 / (-1.0 - t)) - 2.0)) - 2.0));
end

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(N[(8.0 + N[(-8.0 / N[Power[N[(1.0 + t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(N[(4.0 + N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. flip3-- 100.0%

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

   2. associate-*r/ 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(8 - {\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{3}\right)}{\mathsf{fma}\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}, 2 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}, 4\right)}}} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(2 + \frac{-2}{t + 1}\right) \cdot \frac{8 + \frac{-8}{{\left(t + 1\right)}^{3}}}{4 + \frac{4 + \frac{4}{t + 1}}{t + 1}}}} \]

7. Final simplification 100.0%

   \[\leadsto 1 + \frac{1}{\frac{8 + \frac{-8}{{\left(1 + t\right)}^{3}}}{4 + \frac{4 + \frac{4}{1 + t}}{1 + t}} \cdot \left(\frac{-2}{-1 - t} - 2\right) - 2} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;1 - \left(0.16666666666666666 - \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\left(2 \cdot t\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -1.2) (not (<= t 0.58)))
   (-
    1.0
    (-
     0.16666666666666666
     (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)))
   (+ 1.0 (/ 1.0 (- (* (* 2.0 t) (- (/ 2.0 (+ 1.0 t)) 2.0)) 2.0)))))

C

double code(double t) {
	double tmp;
	if ((t <= -1.2) || !(t <= 0.58)) {
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / t) + -0.2222222222222222) / t));
	} else {
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.2d0)) .or. (.not. (t <= 0.58d0))) then
        tmp = 1.0d0 - (0.16666666666666666d0 - (((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t))
    else
        tmp = 1.0d0 + (1.0d0 / (((2.0d0 * t) * ((2.0d0 / (1.0d0 + t)) - 2.0d0)) - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -1.2) || !(t <= 0.58)) {
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / t) + -0.2222222222222222) / t));
	} else {
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -1.2) or not (t <= 0.58):
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / t) + -0.2222222222222222) / t))
	else:
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -1.2) || !(t <= 0.58))
		tmp = Float64(1.0 - Float64(0.16666666666666666 - Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t)));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(2.0 * t) * Float64(Float64(2.0 / Float64(1.0 + t)) - 2.0)) - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -1.2) || ~((t <= 0.58)))
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / t) + -0.2222222222222222) / t));
	else
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((2.0 / (1.0 + t)) - 2.0)) - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -1.2], N[Not[LessEqual[t, 0.58]], $MachinePrecision]], N[(1.0 - N[(0.16666666666666666 - N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(N[(N[(2.0 * t), $MachinePrecision] * N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.19999999999999996 or 0.57999999999999996 < 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 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--r- 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. unpow2 98.3%

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

      8. associate-/r* 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r/ 98.3%

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

      11. div-sub 98.3%

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

      12. sub-neg 98.3%

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

      13. associate-*r/ 98.3%

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

      14. metadata-eval 98.3%

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

      15. distribute-neg-frac 98.3%

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

      16. metadata-eval 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   if -1.19999999999999996 < t < 0.57999999999999996

   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 99.8%

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

      1. add-log-exp 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. log-prod 99.8%

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

      4. metadata-eval 99.8%

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

      5. add-log-exp 99.8%

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

      6. associate-/l/ 99.8%

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

      7. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. +-lft-identity 99.8%

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

      2. distribute-rgt-in 99.8%

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

      3. lft-mult-inverse 99.8%

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

      4. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;1 - \left(0.16666666666666666 - \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\left(2 \cdot t\right) \cdot \left(\frac{2}{1 + t} - 2\right) - 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;1 - \left(0.16666666666666666 - \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.62) (not (<= t 0.44)))
   (-
    1.0
    (-
     0.16666666666666666
     (/ (+ (/ 0.037037037037037035 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 <= -0.62) || !(t <= 0.44)) {
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / 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 <= (-0.62d0)) .or. (.not. (t <= 0.44d0))) then
        tmp = 1.0d0 - (0.16666666666666666d0 - (((0.037037037037037035d0 / 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 <= -0.62) || !(t <= 0.44)) {
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / 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 <= -0.62) or not (t <= 0.44):
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / 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 <= -0.62) || !(t <= 0.44))
		tmp = Float64(1.0 - Float64(0.16666666666666666 - Float64(Float64(Float64(0.037037037037037035 / 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 <= -0.62) || ~((t <= 0.44)))
		tmp = 1.0 - (0.16666666666666666 - (((0.037037037037037035 / 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, -0.62], N[Not[LessEqual[t, 0.44]], $MachinePrecision]], N[(1.0 - N[(0.16666666666666666 - N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $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 < -0.619999999999999996 or 0.440000000000000002 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--r- 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. unpow2 98.3%

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

      8. associate-/r* 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r/ 98.3%

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

      11. div-sub 98.3%

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

      12. sub-neg 98.3%

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

      13. associate-*r/ 98.3%

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

      14. metadata-eval 98.3%

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

      15. distribute-neg-frac 98.3%

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

      16. metadata-eval 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   if -0.619999999999999996 < t < 0.440000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   4. Taylor expanded in t around 0 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;1 - \left(0.16666666666666666 - \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.5) (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 <= -0.5) || !(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 <= (-0.5d0)) .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 <= -0.5) || !(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 <= -0.5) 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 <= -0.5) || !(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 <= -0.5) || ~((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, -0.5], 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 < -0.5 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 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--r- 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. unpow2 98.3%

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

      8. associate-/r* 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r/ 98.3%

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

      11. div-sub 98.3%

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

      12. sub-neg 98.3%

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

      13. associate-*r/ 98.3%

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

      14. metadata-eval 98.3%

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

      15. distribute-neg-frac 98.3%

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

      16. metadata-eval 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   if -0.5 < 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 99.8%

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

   4. Taylor expanded in t around inf 99.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{4 \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in t around 0 99.6%

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

4. Final simplification 99.1%

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

Alternative 5: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-cube-cbrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. add-log-exp 63.3%

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

   2. *-un-lft-identity 63.3%

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

   3. log-prod 63.3%

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

   4. metadata-eval 63.3%

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

   5. add-log-exp 63.3%

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

   6. associate-/l/ 63.3%

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

   7. *-commutative 63.3%

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

6. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right) \cdot \left(2 - {\left(\sqrt[3]{\frac{2}{t}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 63.3%

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

   2. distribute-rgt-in 63.3%

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

   3. lft-mult-inverse 63.3%

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

   4. *-lft-identity 63.3%

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

8. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. associate-*r/ 100.0%

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

   3. unpow2 100.0%

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

   4. add-cube-cbrt 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. rem-cube-cbrt 100.0%

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

    2. associate-/r* 100.0%

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

    3. rem-cube-cbrt 100.0%

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

    4. distribute-lft-in 100.0%

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

    5. *-rgt-identity 100.0%

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

    6. rgt-mult-inverse 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 6: 98.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf 98.3%

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

4. Final simplification 98.3%

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

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--r- 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. unpow2 98.3%

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

      8. associate-/r* 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r/ 98.3%

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

      11. div-sub 98.3%

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

      12. sub-neg 98.3%

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

      13. associate-*r/ 98.3%

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

      14. metadata-eval 98.3%

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

      15. distribute-neg-frac 98.3%

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

      16. metadata-eval 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in t around inf 98.2%

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

   10. Taylor expanded in t around inf 98.2%

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

       1. cancel-sign-sub-inv 98.2%

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

       2. metadata-eval 98.2%

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

       3. associate-*r/ 98.2%

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

       4. metadata-eval 98.2%

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

       5. +-commutative 98.2%

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

   12. Simplified 98.2%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   4. Taylor expanded in t around inf 99.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{4 \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in t around 0 99.6%

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

4. Final simplification 99.0%

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

Alternative 8: 98.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--r- 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. unpow2 98.3%

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

      8. associate-/r* 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r/ 98.3%

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

      11. div-sub 98.3%

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

      12. sub-neg 98.3%

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

      13. associate-*r/ 98.3%

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

      14. metadata-eval 98.3%

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

      15. distribute-neg-frac 98.3%

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

      16. metadata-eval 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in t around inf 98.2%

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

   10. Taylor expanded in t around inf 97.5%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   4. Taylor expanded in t around inf 99.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{4 \cdot t}} \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in t around 0 99.6%

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

4. Final simplification 98.6%

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

Alternative 9: 58.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

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 inf 55.7%

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

   1. +-commutative 55.7%

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

   2. associate--l+ 55.7%

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

   3. +-commutative 55.7%

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

   4. associate--r- 55.7%

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

   5. associate-*r/ 55.7%

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

   6. metadata-eval 55.7%

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

   7. unpow2 55.7%

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

   8. associate-/r* 55.7%

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

   9. metadata-eval 55.7%

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

   10. associate-*r/ 55.7%

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

   11. div-sub 55.7%

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

   12. sub-neg 55.7%

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

   13. associate-*r/ 55.7%

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

   14. metadata-eval 55.7%

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

   15. distribute-neg-frac 55.7%

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

   16. metadata-eval 55.7%

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

5. Simplified 55.7%

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

6. Taylor expanded in t around -inf 47.9%

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

   1. mul-1-neg 47.9%

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

   2. unsub-neg 47.9%

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

   3. sub-neg 47.9%

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

   4. associate-*r/ 47.9%

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

   5. metadata-eval 47.9%

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

   6. metadata-eval 47.9%

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

8. Simplified 47.9%

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

9. Taylor expanded in t around inf 47.4%

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

10. Taylor expanded in t around inf 55.8%

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

11. Final simplification 55.8%

    \[\leadsto 0.8333333333333334 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))))))))