Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 48.6s

Alternatives: 10

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l/ 100.0%

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

   2. div-inv 100.0%

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

   3. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. mul-1-neg 100.0%

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

   5. div-inv 100.0%

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

   6. add-sqr-sqrt 53.1%

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

   7. sqrt-unprod 73.1%

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

   8. frac-times 73.1%

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

   9. metadata-eval 73.1%

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

   10. metadata-eval 73.1%

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

   11. frac-times 73.1%

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

   12. sqrt-unprod 45.0%

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

   13. add-sqr-sqrt 96.9%

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

   14. distribute-neg-in 96.9%

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

   15. metadata-eval 96.9%

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

   16. mul-1-neg 96.9%

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

   17. div-inv 96.9%

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

   18. +-commutative 96.9%

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

   19. associate-/l* 96.9%

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

   20. associate-/r* 96.9%

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

   21. +-commutative 96.9%

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

   22. distribute-lft-in 96.9%

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

6. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. *-inverses 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 73.4%

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

   2. expm1-undefine 73.4%

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

   3. distribute-lft-in 73.4%

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

   4. distribute-lft-in 73.4%

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

   5. distribute-rgt-in 73.4%

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

   6. *-un-lft-identity 73.4%

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

   7. lft-mult-inverse 73.4%

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

10. Applied egg-rr 73.4%

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

    1. sub-neg 73.4%

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

    2. metadata-eval 73.4%

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

    3. +-commutative 73.4%

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

    4. log1p-undefine 73.4%

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

    5. rem-exp-log 100.0%

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

    6. +-commutative 100.0%

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

    7. associate-+r+ 100.0%

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

    8. metadata-eval 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;t\_1 \leq 0.5:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\right) \cdot \left(t\_1 - 2\right)}{2 + \left(t \cdot \left(2 + t \cdot -2\right)\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))
   (if (<= t_1 0.5)
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t))
     (/
      (- 1.0 (* (+ 2.0 (/ (/ 2.0 t) (+ -1.0 (/ -1.0 t)))) (- t_1 2.0)))
      (+ 2.0 (* (* t (+ 2.0 (* t -2.0))) (* 2.0 t)))))))

C

double code(double t) {
	double t_1 = (2.0 / t) / (1.0 + (1.0 / t));
	double tmp;
	if (t_1 <= 0.5) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = (1.0 - ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t_1 - 2.0))) / (2.0 + ((t * (2.0 + (t * -2.0))) * (2.0 * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / t) / (1.0d0 + (1.0d0 / t))
    if (t_1 <= 0.5d0) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = (1.0d0 - ((2.0d0 + ((2.0d0 / t) / ((-1.0d0) + ((-1.0d0) / t)))) * (t_1 - 2.0d0))) / (2.0d0 + ((t * (2.0d0 + (t * (-2.0d0)))) * (2.0d0 * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (2.0 / t) / (1.0 + (1.0 / t));
	double tmp;
	if (t_1 <= 0.5) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = (1.0 - ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t_1 - 2.0))) / (2.0 + ((t * (2.0 + (t * -2.0))) * (2.0 * t)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (2.0 / t) / (1.0 + (1.0 / t))
	tmp = 0
	if t_1 <= 0.5:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = (1.0 - ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t_1 - 2.0))) / (2.0 + ((t * (2.0 + (t * -2.0))) * (2.0 * t)))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))
	tmp = 0.0
	if (t_1 <= 0.5)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(2.0 + Float64(Float64(2.0 / t) / Float64(-1.0 + Float64(-1.0 / t)))) * Float64(t_1 - 2.0))) / Float64(2.0 + Float64(Float64(t * Float64(2.0 + Float64(t * -2.0))) * Float64(2.0 * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (2.0 / t) / (1.0 + (1.0 / t));
	tmp = 0.0;
	if (t_1 <= 0.5)
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = (1.0 - ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t_1 - 2.0))) / (2.0 + ((t * (2.0 + (t * -2.0))) * (2.0 * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.5], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(t * N[(2.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.7%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 98.7%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 98.7%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. mul-1-neg 98.7%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. unsub-neg 98.7%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. sub-neg 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. associate-*r/ 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      8. distribute-neg-frac 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      9. metadata-eval 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 98.7%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 98.8%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   8. Simplified 98.8%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

   9. Taylor expanded in t around -inf 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

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

       3. mul-1-neg 99.0%

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

       4. unsub-neg 99.0%

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

       5. associate-*r/ 99.0%

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

       6. metadata-eval 99.0%

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

   11. Simplified 99.0%

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

   if 0.5 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.5:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)}{2 + \left(t \cdot \left(2 + t \cdot -2\right)\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (2.0 + (2.0 / (-1.0 - t)));
	return (1.0 + t_1) / (2.0 + 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)))) * (2.0d0 + (2.0d0 / ((-1.0d0) - t)))
    code = (1.0d0 + t_1) / (2.0d0 + t_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l/ 100.0%

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

   2. div-inv 100.0%

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

   3. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. mul-1-neg 100.0%

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

   5. div-inv 100.0%

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

   6. add-sqr-sqrt 53.1%

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

   7. sqrt-unprod 73.1%

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

   8. frac-times 73.1%

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

   9. metadata-eval 73.1%

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

   10. metadata-eval 73.1%

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

   11. frac-times 73.1%

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

   12. sqrt-unprod 45.0%

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

   13. add-sqr-sqrt 96.9%

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

   14. distribute-neg-in 96.9%

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

   15. metadata-eval 96.9%

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

   16. mul-1-neg 96.9%

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

   17. div-inv 96.9%

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

   18. +-commutative 96.9%

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

   19. associate-/l* 96.9%

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

   20. associate-/r* 96.9%

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

   21. +-commutative 96.9%

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

   22. distribute-lft-in 96.9%

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

6. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. *-inverses 100.0%

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

8. Simplified 100.0%

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

   1. cancel-sign-sub-inv 100.0%

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

   2. un-div-inv 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. lft-mult-inverse 100.0%

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

10. Applied egg-rr 100.0%

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

    1. metadata-eval 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. distribute-neg-frac2 100.0%

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

    4. neg-sub0 100.0%

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

    5. +-commutative 100.0%

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

    6. associate--r+ 100.0%

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

    7. metadata-eval 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 4: 99.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.5:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\right) \cdot \left(t \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.5)
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   (/ 1.0 (+ 2.0 (* (+ 2.0 (/ (/ 2.0 t) (+ -1.0 (/ -1.0 t)))) (* t -2.0))))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.5) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 1.0 / (2.0 + ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t * -2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 + (1.0d0 / t))) <= 0.5d0) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = 1.0d0 / (2.0d0 + ((2.0d0 + ((2.0d0 / t) / ((-1.0d0) + ((-1.0d0) / t)))) * (t * (-2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.5) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 1.0 / (2.0 + ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t * -2.0)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 0.5:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = 1.0 / (2.0 + ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t * -2.0)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.5)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(2.0 + Float64(Float64(2.0 / t) / Float64(-1.0 + Float64(-1.0 / t)))) * Float64(t * -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.5)
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = 1.0 / (2.0 + ((2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))) * (t * -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], 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[(2.0 + N[(N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.7%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 98.7%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 98.7%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. mul-1-neg 98.7%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. unsub-neg 98.7%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. sub-neg 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. associate-*r/ 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      8. distribute-neg-frac 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      9. metadata-eval 98.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 98.7%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 98.8%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 98.8%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   8. Simplified 98.8%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

   9. Taylor expanded in t around -inf 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

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

       3. mul-1-neg 99.0%

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

       4. unsub-neg 99.0%

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

       5. associate-*r/ 99.0%

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

       6. metadata-eval 99.0%

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

   11. Simplified 99.0%

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

   if 0.5 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

      1. frac-2neg 98.6%

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

      2. div-inv 98.6%

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

      3. div-inv 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

      5. mul-1-neg 98.6%

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

      6. div-inv 98.6%

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

      7. add-sqr-sqrt 57.5%

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

      8. sqrt-unprod 43.2%

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

      9. frac-times 43.2%

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

      10. metadata-eval 43.2%

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

      11. metadata-eval 43.2%

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

      12. frac-times 43.2%

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

      13. sqrt-unprod 40.4%

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

      14. add-sqr-sqrt 96.8%

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

      15. div-inv 96.8%

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

      16. distribute-neg-in 96.8%

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

      17. metadata-eval 96.8%

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

      18. mul-1-neg 96.8%

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

      19. div-inv 96.8%

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

      20. +-commutative 96.8%

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

   5. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.5:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\right) \cdot \left(t \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.68)
     0.5
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.6%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 97.6%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 97.6%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. mul-1-neg 97.6%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. unsub-neg 97.6%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. sub-neg 97.6%

         \[\leadsto \frac{5 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. associate-*r/ 97.6%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 97.6%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      8. distribute-neg-frac 97.6%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      9. metadata-eval 97.6%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 97.6%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 97.7%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}\right)} \]

      2. unsub-neg 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      3. mul-1-neg 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}\right)} \]

      4. unsub-neg 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}\right)} \]

      5. sub-neg 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}\right)} \]

      6. associate-*r/ 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}\right)} \]

      7. metadata-eval 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}\right)} \]

      8. distribute-neg-frac 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}\right)} \]

      9. metadata-eval 97.7%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \left(4 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}\right)} \]

   8. Simplified 97.7%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{2 + \color{blue}{\left(4 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}\right)}} \]

   9. Taylor expanded in t around -inf 98.0%

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

       1. mul-1-neg 98.0%

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

       2. unsub-neg 98.0%

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

       3. mul-1-neg 98.0%

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

       4. unsub-neg 98.0%

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

       5. associate-*r/ 98.0%

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

       6. metadata-eval 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right) \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.23)
     0.5
     (+
      0.8333333333333334
      (* (+ 0.2222222222222222 (/ -0.037037037037037035 t)) (/ -1.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((0.2222222222222222 + (-0.037037037037037035 / t)) * (-1.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.23d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) * ((-1.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((0.2222222222222222 + (-0.037037037037037035 / t)) * (-1.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.23:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + ((0.2222222222222222 + (-0.037037037037037035 / t)) * (-1.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) * Float64(-1.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + ((0.2222222222222222 + (-0.037037037037037035 / t)) * (-1.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.23], 0.5, N[(0.8333333333333334 + N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if -0.48999999999999999 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

   if 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

      3. sub-neg 97.3%

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

      4. associate-*r/ 97.3%

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

      5. metadata-eval 97.3%

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

      6. distribute-neg-frac 97.3%

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

      7. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in t around inf 97.7%

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

      1. sub-neg 97.7%

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

      2. associate-+l+ 97.7%

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

      3. sub-neg 97.7%

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

      4. unpow2 97.7%

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

      5. associate-/r* 97.7%

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

      6. metadata-eval 97.7%

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

      7. associate-*r/ 97.7%

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

      8. associate-*r/ 97.7%

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

      9. metadata-eval 97.7%

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

      10. div-sub 97.7%

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

      11. remove-double-neg 97.7%

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

      12. mul-1-neg 97.7%

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

      13. sub-neg 97.7%

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

      14. distribute-lft-in 97.7%

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

      15. neg-mul-1 97.7%

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

      16. metadata-eval 97.7%

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

      17. metadata-eval 97.7%

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

      18. +-commutative 97.7%

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

      19. sub-neg 97.7%

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

   8. Simplified 97.7%

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

      1. div-inv 97.7%

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

      2. +-commutative 97.7%

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

   10. Applied egg-rr 97.7%

       \[\leadsto 0.8333333333333334 - \color{blue}{\left(\frac{-0.037037037037037035}{t} + 0.2222222222222222\right) \cdot \frac{1}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(0.2222222222222222 + \frac{-0.037037037037037035}{t}\right) \cdot \frac{-1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.23)
     0.5
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.23d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.23:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.23], 0.5, N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if -0.48999999999999999 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

   if 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

      3. sub-neg 97.3%

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

      4. associate-*r/ 97.3%

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

      5. metadata-eval 97.3%

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

      6. distribute-neg-frac 97.3%

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

      7. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in t around inf 97.7%

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

      1. sub-neg 97.7%

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

      2. associate-+l+ 97.7%

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

      3. sub-neg 97.7%

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

      4. unpow2 97.7%

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

      5. associate-/r* 97.7%

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

      6. metadata-eval 97.7%

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

      7. associate-*r/ 97.7%

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

      8. associate-*r/ 97.7%

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

      9. metadata-eval 97.7%

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

      10. div-sub 97.7%

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

      11. remove-double-neg 97.7%

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

      12. mul-1-neg 97.7%

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

      13. sub-neg 97.7%

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

      14. distribute-lft-in 97.7%

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

      15. neg-mul-1 97.7%

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

      16. metadata-eval 97.7%

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

      17. metadata-eval 97.7%

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

      18. +-commutative 97.7%

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

      19. sub-neg 97.7%

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

   8. Simplified 97.7%

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

Alternative 8: 99.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. associate-*r/ 98.6%

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

      5. metadata-eval 98.6%

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

      6. distribute-neg-frac 98.6%

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

      7. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   8. Simplified 98.7%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

4. Final simplification 98.7%

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

Alternative 9: 98.6% accurate, 4.6× 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%

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

   3. Taylor expanded in t around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. associate-*r/ 98.6%

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

      5. metadata-eval 98.6%

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

      6. distribute-neg-frac 98.6%

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

      7. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around inf 97.4%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.6%

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

   4. Taylor expanded in t around 0 98.7%

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

Alternative 10: 58.9% accurate, 51.0× 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%

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

3. Taylor expanded in t around 0 54.2%

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

4. Taylor expanded in t around 0 55.8%

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

Reproduce

herbie shell --seed 2024084 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))