Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

Alternatives: 11

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = -1.0 - (1.0 / t);
	t_2 = 2.0 + ((2.0 / t) / t_1);
	tmp = (1.0 + (t_2 * (2.0 + (2.0 / (t * t_1))))) / (2.0 + (t_2 * (2.0 + (2.0 / (-1.0 - t)))));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(-1.0 - N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$2 * N[(2.0 + N[(2.0 / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$2 * N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $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%

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

   2. div-inv 100.0%

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

   3. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. mul-1-neg 100.0%

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

   5. div-inv 100.0%

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

   6. add-sqr-sqrt 41.0%

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

   7. sqrt-unprod 67.8%

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

   8. frac-times 67.8%

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

   9. metadata-eval 67.8%

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

   10. metadata-eval 67.8%

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

   11. frac-times 67.8%

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

   12. sqrt-unprod 58.1%

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

   13. add-sqr-sqrt 97.9%

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

   14. div-inv 97.9%

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

   15. distribute-neg-in 97.9%

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

   16. metadata-eval 97.9%

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

   17. mul-1-neg 97.9%

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

   18. div-inv 97.9%

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

   19. +-commutative 97.9%

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

   20. *-un-lft-identity 97.9%

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

   21. associate-/r* 97.9%

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

   22. +-commutative 97.9%

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

   23. distribute-lft-in 97.9%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-inverses 100.0%

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

8. Simplified 100.0%

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

   1. un-div-inv 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

10. Applied egg-rr 100.0%

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

    1. associate-/l/ 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 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%

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

   2. div-inv 100.0%

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

   3. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. mul-1-neg 100.0%

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

   5. div-inv 100.0%

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

   6. add-sqr-sqrt 41.0%

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

   7. sqrt-unprod 67.8%

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

   8. frac-times 67.8%

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

   9. metadata-eval 67.8%

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

   10. metadata-eval 67.8%

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

   11. frac-times 67.8%

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

   12. sqrt-unprod 58.1%

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

   13. add-sqr-sqrt 97.9%

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

   14. div-inv 97.9%

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

   15. distribute-neg-in 97.9%

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

   16. metadata-eval 97.9%

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

   17. mul-1-neg 97.9%

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

   18. div-inv 97.9%

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

   19. +-commutative 97.9%

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

   20. *-un-lft-identity 97.9%

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

   21. associate-/r* 97.9%

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

   22. +-commutative 97.9%

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

   23. distribute-lft-in 97.9%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-inverses 100.0%

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

8. Simplified 100.0%

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

   1. un-div-inv 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. inv-pow 100.0%

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

   5. pow-plus 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. div-inv 100.0%

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

   3. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. mul-1-neg 100.0%

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

   5. div-inv 100.0%

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

   6. add-sqr-sqrt 41.0%

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

   7. sqrt-unprod 67.8%

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

   8. frac-times 67.8%

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

   9. metadata-eval 67.8%

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

   10. metadata-eval 67.8%

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

   11. frac-times 67.8%

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

   12. sqrt-unprod 58.1%

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

   13. add-sqr-sqrt 97.9%

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

   14. div-inv 97.9%

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

   15. distribute-neg-in 97.9%

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

   16. metadata-eval 97.9%

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

   17. mul-1-neg 97.9%

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

   18. div-inv 97.9%

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

   19. +-commutative 97.9%

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

   20. *-un-lft-identity 97.9%

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

   21. associate-/r* 97.9%

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

   22. +-commutative 97.9%

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

   23. distribute-lft-in 97.9%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-inverses 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in t around inf 97.5%

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

    1. sub-neg 97.5%

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

    2. associate-*r/ 97.5%

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

    3. metadata-eval 97.5%

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

    4. distribute-neg-frac 97.5%

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

    5. metadata-eval 97.5%

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

11. Simplified 97.5%

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

12. Final simplification 97.5%

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

Alternative 4: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 + \frac{1}{t \cdot \left(\frac{\frac{\frac{0.003472222222222222 \cdot \frac{-1}{t} - 0.020833333333333332}{t} - 0.125}{t} - 0.75}{t} - 4.5\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  0.8333333333333334
  (/
   1.0
   (*
    t
    (-
     (/
      (-
       (/
        (-
         (/ (- (* 0.003472222222222222 (/ -1.0 t)) 0.020833333333333332) t)
         0.125)
        t)
       0.75)
      t)
     4.5)))))

C

double code(double t) {
	return 0.8333333333333334 + (1.0 / (t * ((((((((0.003472222222222222 * (-1.0 / t)) - 0.020833333333333332) / t) - 0.125) / t) - 0.75) / t) - 4.5)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0 + (1.0d0 / (t * ((((((((0.003472222222222222d0 * ((-1.0d0) / t)) - 0.020833333333333332d0) / t) - 0.125d0) / t) - 0.75d0) / t) - 4.5d0)))
end function

Java

public static double code(double t) {
	return 0.8333333333333334 + (1.0 / (t * ((((((((0.003472222222222222 * (-1.0 / t)) - 0.020833333333333332) / t) - 0.125) / t) - 0.75) / t) - 4.5)));
}

Python

def code(t):
	return 0.8333333333333334 + (1.0 / (t * ((((((((0.003472222222222222 * (-1.0 / t)) - 0.020833333333333332) / t) - 0.125) / t) - 0.75) / t) - 4.5)))

Julia

function code(t)
	return Float64(0.8333333333333334 + Float64(1.0 / Float64(t * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.003472222222222222 * Float64(-1.0 / t)) - 0.020833333333333332) / t) - 0.125) / t) - 0.75) / t) - 4.5))))
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334 + (1.0 / (t * ((((((((0.003472222222222222 * (-1.0 / t)) - 0.020833333333333332) / t) - 0.125) / t) - 0.75) / t) - 4.5)));
end

Wolfram

code[t_] := N[(0.8333333333333334 + N[(1.0 / N[(t * N[(N[(N[(N[(N[(N[(N[(N[(0.003472222222222222 * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision] - 0.020833333333333332), $MachinePrecision] / t), $MachinePrecision] - 0.125), $MachinePrecision] / t), $MachinePrecision] - 0.75), $MachinePrecision] / t), $MachinePrecision] - 4.5), $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. Taylor expanded in t around -inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. unsub-neg 44.2%

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

   3. sub-neg 44.2%

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

   4. associate-*r/ 44.2%

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

   5. metadata-eval 44.2%

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

   6. distribute-neg-frac 44.2%

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

   7. metadata-eval 44.2%

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

5. Simplified 44.2%

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

   1. clear-num 44.2%

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

   2. inv-pow 44.2%

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

7. Applied egg-rr 44.2%

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

   1. unpow-1 44.2%

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

9. Simplified 44.2%

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

10. Taylor expanded in t around -inf 53.2%

    \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + -1 \cdot \frac{-1 \cdot \frac{0.020833333333333332 + 0.003472222222222222 \cdot \frac{1}{t}}{t} - 0.125}{t}}{t} - 4.5\right)\right)}} \]

11. Final simplification 53.2%

    \[\leadsto 0.8333333333333334 + \frac{1}{t \cdot \left(\frac{\frac{\frac{0.003472222222222222 \cdot \frac{-1}{t} - 0.020833333333333332}{t} - 0.125}{t} - 0.75}{t} - 4.5\right)} \]
12. Add Preprocessing

Alternative 5: 60.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.22 \lor \neg \left(t \leq 0.09\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + t \cdot -8\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.22) (not (<= t 0.09)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (+ 0.8333333333333334 (* t -8.0))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.22) || !(t <= 0.09)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.8333333333333334 + (t * -8.0);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.22d0)) .or. (.not. (t <= 0.09d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = 0.8333333333333334d0 + (t * (-8.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.22) || !(t <= 0.09)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.8333333333333334 + (t * -8.0);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.22) or not (t <= 0.09):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = 0.8333333333333334 + (t * -8.0)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.22) || !(t <= 0.09))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = Float64(0.8333333333333334 + Float64(t * -8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.22) || ~((t <= 0.09)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = 0.8333333333333334 + (t * -8.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.22], N[Not[LessEqual[t, 0.09]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(t * -8.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.220000000000000001 or 0.089999999999999997 < 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 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. associate-*r/ 98.4%

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

      5. metadata-eval 98.4%

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

      6. distribute-neg-frac 98.4%

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

      7. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.220000000000000001 < t < 0.089999999999999997

   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 4.0%

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

      1. mul-1-neg 4.0%

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

      2. unsub-neg 4.0%

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

      3. sub-neg 4.0%

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

      4. associate-*r/ 4.0%

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

      5. metadata-eval 4.0%

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

      6. distribute-neg-frac 4.0%

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

      7. metadata-eval 4.0%

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

   5. Simplified 4.0%

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

      1. clear-num 4.0%

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

      2. inv-pow 4.0%

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

   7. Applied egg-rr 4.0%

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

      1. unpow-1 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in t around -inf 19.7%

       \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)\right)}} \]

   11. Taylor expanded in t around 0 19.7%

       \[\leadsto \color{blue}{0.8333333333333334 + -8 \cdot t} \]
   12. Step-by-step derivation

       1. *-commutative 19.7%

          \[\leadsto 0.8333333333333334 + \color{blue}{t \cdot -8} \]

   13. Simplified 19.7%

       \[\leadsto \color{blue}{0.8333333333333334 + t \cdot -8} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.22 \lor \neg \left(t \leq 0.09\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + t \cdot -8\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 + \frac{1}{t \cdot \left(\frac{0.125 \cdot \frac{-1}{t} - 0.75}{t} - 4.5\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  0.8333333333333334
  (/ 1.0 (* t (- (/ (- (* 0.125 (/ -1.0 t)) 0.75) t) 4.5)))))

C

double code(double t) {
	return 0.8333333333333334 + (1.0 / (t * ((((0.125 * (-1.0 / t)) - 0.75) / t) - 4.5)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0 + (1.0d0 / (t * ((((0.125d0 * ((-1.0d0) / t)) - 0.75d0) / t) - 4.5d0)))
end function

Java

public static double code(double t) {
	return 0.8333333333333334 + (1.0 / (t * ((((0.125 * (-1.0 / t)) - 0.75) / t) - 4.5)));
}

Python

def code(t):
	return 0.8333333333333334 + (1.0 / (t * ((((0.125 * (-1.0 / t)) - 0.75) / t) - 4.5)))

Julia

function code(t)
	return Float64(0.8333333333333334 + Float64(1.0 / Float64(t * Float64(Float64(Float64(Float64(0.125 * Float64(-1.0 / t)) - 0.75) / t) - 4.5))))
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334 + (1.0 / (t * ((((0.125 * (-1.0 / t)) - 0.75) / t) - 4.5)));
end

Wolfram

code[t_] := N[(0.8333333333333334 + N[(1.0 / N[(t * N[(N[(N[(N[(0.125 * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision] - 0.75), $MachinePrecision] / t), $MachinePrecision] - 4.5), $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. Taylor expanded in t around -inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. unsub-neg 44.2%

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

   3. sub-neg 44.2%

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

   4. associate-*r/ 44.2%

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

   5. metadata-eval 44.2%

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

   6. distribute-neg-frac 44.2%

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

   7. metadata-eval 44.2%

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

5. Simplified 44.2%

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

   1. clear-num 44.2%

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

   2. inv-pow 44.2%

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

7. Applied egg-rr 44.2%

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

   1. unpow-1 44.2%

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

9. Simplified 44.2%

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

10. Taylor expanded in t around -inf 53.2%

    \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)\right)}} \]

11. Final simplification 53.2%

    \[\leadsto 0.8333333333333334 + \frac{1}{t \cdot \left(\frac{0.125 \cdot \frac{-1}{t} - 0.75}{t} - 4.5\right)} \]
12. Add Preprocessing

Alternative 7: 59.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1.8333333333333333 + \frac{1}{t \cdot \left(-4.5 + \frac{-0.75 + \frac{-0.125}{t}}{t}\right)}\right) \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  -1.0
  (+ 1.8333333333333333 (/ 1.0 (* t (+ -4.5 (/ (+ -0.75 (/ -0.125 t)) t)))))))

C

double code(double t) {
	return -1.0 + (1.8333333333333333 + (1.0 / (t * (-4.5 + ((-0.75 + (-0.125 / t)) / t)))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (-1.0d0) + (1.8333333333333333d0 + (1.0d0 / (t * ((-4.5d0) + (((-0.75d0) + ((-0.125d0) / t)) / t)))))
end function

Java

public static double code(double t) {
	return -1.0 + (1.8333333333333333 + (1.0 / (t * (-4.5 + ((-0.75 + (-0.125 / t)) / t)))));
}

Python

def code(t):
	return -1.0 + (1.8333333333333333 + (1.0 / (t * (-4.5 + ((-0.75 + (-0.125 / t)) / t)))))

Julia

function code(t)
	return Float64(-1.0 + Float64(1.8333333333333333 + Float64(1.0 / Float64(t * Float64(-4.5 + Float64(Float64(-0.75 + Float64(-0.125 / t)) / t))))))
end

MATLAB

function tmp = code(t)
	tmp = -1.0 + (1.8333333333333333 + (1.0 / (t * (-4.5 + ((-0.75 + (-0.125 / t)) / t)))));
end

Wolfram

code[t_] := N[(-1.0 + N[(1.8333333333333333 + N[(1.0 / N[(t * N[(-4.5 + N[(N[(-0.75 + N[(-0.125 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in t around -inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. unsub-neg 44.2%

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

   3. sub-neg 44.2%

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

   4. associate-*r/ 44.2%

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

   5. metadata-eval 44.2%

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

   6. distribute-neg-frac 44.2%

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

   7. metadata-eval 44.2%

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

5. Simplified 44.2%

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

   1. clear-num 44.2%

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

   2. inv-pow 44.2%

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

7. Applied egg-rr 44.2%

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

   1. unpow-1 44.2%

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

9. Simplified 44.2%

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

10. Taylor expanded in t around -inf 53.2%

    \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)\right)}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 52.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{1}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)\right)}\right)\right)} \]

    2. associate-/r* 52.5%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \color{blue}{\frac{\frac{1}{-1}}{t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)}}\right)\right) \]

    3. metadata-eval 52.5%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{\color{blue}{-1}}{t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)}\right)\right) \]

    4. fmm-def 52.5%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{-1}{t \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t}, -4.5\right)}}\right)\right) \]

    5. un-div-inv 52.5%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{-1}{t \cdot \mathsf{fma}\left(-1, \frac{0.75 + \color{blue}{\frac{0.125}{t}}}{t}, -4.5\right)}\right)\right) \]

    6. metadata-eval 52.5%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{-1}{t \cdot \mathsf{fma}\left(-1, \frac{0.75 + \frac{0.125}{t}}{t}, \color{blue}{-4.5}\right)}\right)\right) \]

12. Applied egg-rr 52.5%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{-1}{t \cdot \mathsf{fma}\left(-1, \frac{0.75 + \frac{0.125}{t}}{t}, -4.5\right)}\right)\right)} \]
13. Step-by-step derivation

    1. expm1-undefine 52.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{-1}{t \cdot \mathsf{fma}\left(-1, \frac{0.75 + \frac{0.125}{t}}{t}, -4.5\right)}\right)} - 1} \]

    2. sub-neg 52.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{-1}{t \cdot \mathsf{fma}\left(-1, \frac{0.75 + \frac{0.125}{t}}{t}, -4.5\right)}\right)} + \left(-1\right)} \]

14. Simplified 52.5%

    \[\leadsto \color{blue}{\left(1.8333333333333333 + \frac{1}{t \cdot \left(-4.5 + \frac{-0.75 + \frac{-0.125}{t}}{t}\right)}\right) + -1} \]

15. Final simplification 52.5%

    \[\leadsto -1 + \left(1.8333333333333333 + \frac{1}{t \cdot \left(-4.5 + \frac{-0.75 + \frac{-0.125}{t}}{t}\right)}\right) \]
16. Add Preprocessing

Alternative 8: 60.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.17 \lor \neg \left(t \leq 0.165\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + t \cdot -8\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.17) (not (<= t 0.165)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ 0.8333333333333334 (* t -8.0))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.17) || !(t <= 0.165)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.8333333333333334 + (t * -8.0);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.17d0)) .or. (.not. (t <= 0.165d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.8333333333333334d0 + (t * (-8.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.17) || !(t <= 0.165)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.8333333333333334 + (t * -8.0);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.17) or not (t <= 0.165):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.8333333333333334 + (t * -8.0)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.17) || !(t <= 0.165))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(0.8333333333333334 + Float64(t * -8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.17) || ~((t <= 0.165)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.8333333333333334 + (t * -8.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.17], N[Not[LessEqual[t, 0.165]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(t * -8.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.170000000000000012 or 0.165000000000000008 < 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 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   5. Simplified 98.0%

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

   if -0.170000000000000012 < t < 0.165000000000000008

   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 4.0%

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

      1. mul-1-neg 4.0%

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

      2. unsub-neg 4.0%

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

      3. sub-neg 4.0%

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

      4. associate-*r/ 4.0%

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

      5. metadata-eval 4.0%

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

      6. distribute-neg-frac 4.0%

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

      7. metadata-eval 4.0%

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

   5. Simplified 4.0%

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

      1. clear-num 4.0%

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

      2. inv-pow 4.0%

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

   7. Applied egg-rr 4.0%

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

      1. unpow-1 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in t around -inf 19.7%

       \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)\right)}} \]

   11. Taylor expanded in t around 0 19.7%

       \[\leadsto \color{blue}{0.8333333333333334 + -8 \cdot t} \]
   12. Step-by-step derivation

       1. *-commutative 19.7%

          \[\leadsto 0.8333333333333334 + \color{blue}{t \cdot -8} \]

   13. Simplified 19.7%

       \[\leadsto \color{blue}{0.8333333333333334 + t \cdot -8} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.17 \lor \neg \left(t \leq 0.165\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + t \cdot -8\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \left(4 - \frac{8}{t}\right)}{6 - \frac{8}{t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/ (+ 1.0 (- 4.0 (/ 8.0 t))) (- 6.0 (/ 8.0 t))))

C

double code(double t) {
	return (1.0 + (4.0 - (8.0 / t))) / (6.0 - (8.0 / t));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + (4.0d0 - (8.0d0 / t))) / (6.0d0 - (8.0d0 / t))
end function

Java

public static double code(double t) {
	return (1.0 + (4.0 - (8.0 / t))) / (6.0 - (8.0 / t));
}

Python

def code(t):
	return (1.0 + (4.0 - (8.0 / t))) / (6.0 - (8.0 / t))

Julia

function code(t)
	return Float64(Float64(1.0 + Float64(4.0 - Float64(8.0 / t))) / Float64(6.0 - Float64(8.0 / t)))
end

MATLAB

function tmp = code(t)
	tmp = (1.0 + (4.0 - (8.0 / t))) / (6.0 - (8.0 / t));
end

Wolfram

code[t_] := N[(N[(1.0 + N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 - N[(8.0 / t), $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. Taylor expanded in t around inf 43.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
4. Step-by-step derivation

   1. associate-*r/ 43.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

   2. metadata-eval 43.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \frac{\color{blue}{8}}{t}} \]

5. Simplified 43.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - \frac{8}{t}}} \]

6. Taylor expanded in t around inf 52.4%

   \[\leadsto \frac{1 + \color{blue}{\left(4 - 8 \cdot \frac{1}{t}\right)}}{6 - \frac{8}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ 52.4%

      \[\leadsto \frac{1 + \left(4 - \color{blue}{\frac{8 \cdot 1}{t}}\right)}{6 - \frac{8}{t}} \]

   2. metadata-eval 52.4%

      \[\leadsto \frac{1 + \left(4 - \frac{\color{blue}{8}}{t}\right)}{6 - \frac{8}{t}} \]

8. Simplified 52.4%

   \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8}{t}\right)}}{6 - \frac{8}{t}} \]
9. Add Preprocessing

Alternative 10: 58.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(5 + \frac{-8}{t}\right) \cdot \frac{1}{6 + \frac{-8}{t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (* (+ 5.0 (/ -8.0 t)) (/ 1.0 (+ 6.0 (/ -8.0 t)))))

C

double code(double t) {
	return (5.0 + (-8.0 / t)) * (1.0 / (6.0 + (-8.0 / t)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (5.0d0 + ((-8.0d0) / t)) * (1.0d0 / (6.0d0 + ((-8.0d0) / t)))
end function

Java

public static double code(double t) {
	return (5.0 + (-8.0 / t)) * (1.0 / (6.0 + (-8.0 / t)));
}

Python

def code(t):
	return (5.0 + (-8.0 / t)) * (1.0 / (6.0 + (-8.0 / t)))

Julia

function code(t)
	return Float64(Float64(5.0 + Float64(-8.0 / t)) * Float64(1.0 / Float64(6.0 + Float64(-8.0 / t))))
end

MATLAB

function tmp = code(t)
	tmp = (5.0 + (-8.0 / t)) * (1.0 / (6.0 + (-8.0 / t)));
end

Wolfram

code[t_] := N[(N[(5.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(6.0 + N[(-8.0 / t), $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. Taylor expanded in t around inf 43.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
4. Step-by-step derivation

   1. associate-*r/ 43.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

   2. metadata-eval 43.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \frac{\color{blue}{8}}{t}} \]

5. Simplified 43.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - \frac{8}{t}}} \]

6. Taylor expanded in t around inf 52.4%

   \[\leadsto \frac{1 + \color{blue}{\left(4 - 8 \cdot \frac{1}{t}\right)}}{6 - \frac{8}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ 52.4%

      \[\leadsto \frac{1 + \left(4 - \color{blue}{\frac{8 \cdot 1}{t}}\right)}{6 - \frac{8}{t}} \]

   2. metadata-eval 52.4%

      \[\leadsto \frac{1 + \left(4 - \frac{\color{blue}{8}}{t}\right)}{6 - \frac{8}{t}} \]

8. Simplified 52.4%

   \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8}{t}\right)}}{6 - \frac{8}{t}} \]
9. Step-by-step derivation

   1. div-inv 51.7%

      \[\leadsto \color{blue}{\left(1 + \left(4 - \frac{8}{t}\right)\right) \cdot \frac{1}{6 - \frac{8}{t}}} \]

   2. associate-+r- 51.7%

      \[\leadsto \color{blue}{\left(\left(1 + 4\right) - \frac{8}{t}\right)} \cdot \frac{1}{6 - \frac{8}{t}} \]

   3. sub-neg 51.7%

      \[\leadsto \color{blue}{\left(\left(1 + 4\right) + \left(-\frac{8}{t}\right)\right)} \cdot \frac{1}{6 - \frac{8}{t}} \]

   4. metadata-eval 51.7%

      \[\leadsto \left(\color{blue}{5} + \left(-\frac{8}{t}\right)\right) \cdot \frac{1}{6 - \frac{8}{t}} \]

   5. distribute-neg-frac 51.7%

      \[\leadsto \left(5 + \color{blue}{\frac{-8}{t}}\right) \cdot \frac{1}{6 - \frac{8}{t}} \]

   6. metadata-eval 51.7%

      \[\leadsto \left(5 + \frac{\color{blue}{-8}}{t}\right) \cdot \frac{1}{6 - \frac{8}{t}} \]

   7. sub-neg 51.7%

      \[\leadsto \left(5 + \frac{-8}{t}\right) \cdot \frac{1}{\color{blue}{6 + \left(-\frac{8}{t}\right)}} \]

   8. distribute-neg-frac 51.7%

      \[\leadsto \left(5 + \frac{-8}{t}\right) \cdot \frac{1}{6 + \color{blue}{\frac{-8}{t}}} \]

   9. metadata-eval 51.7%

      \[\leadsto \left(5 + \frac{-8}{t}\right) \cdot \frac{1}{6 + \frac{\color{blue}{-8}}{t}} \]

10. Applied egg-rr 51.7%

    \[\leadsto \color{blue}{\left(5 + \frac{-8}{t}\right) \cdot \frac{1}{6 + \frac{-8}{t}}} \]
11. Add Preprocessing

Alternative 11: 11.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 + t \cdot -8 \end{array} \]

FPCore

(FPCore (t) :precision binary64 (+ 0.8333333333333334 (* t -8.0)))

C

double code(double t) {
	return 0.8333333333333334 + (t * -8.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0 + (t * (-8.0d0))
end function

Java

public static double code(double t) {
	return 0.8333333333333334 + (t * -8.0);
}

Python

def code(t):
	return 0.8333333333333334 + (t * -8.0)

Julia

function code(t)
	return Float64(0.8333333333333334 + Float64(t * -8.0))
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334 + (t * -8.0);
end

Wolfram

code[t_] := N[(0.8333333333333334 + N[(t * -8.0), $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. Taylor expanded in t around -inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. unsub-neg 44.2%

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

   3. sub-neg 44.2%

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

   4. associate-*r/ 44.2%

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

   5. metadata-eval 44.2%

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

   6. distribute-neg-frac 44.2%

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

   7. metadata-eval 44.2%

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

5. Simplified 44.2%

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

   1. clear-num 44.2%

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

   2. inv-pow 44.2%

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

7. Applied egg-rr 44.2%

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

   1. unpow-1 44.2%

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

9. Simplified 44.2%

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

10. Taylor expanded in t around -inf 53.2%

    \[\leadsto 0.8333333333333334 - \frac{1}{\color{blue}{-1 \cdot \left(t \cdot \left(-1 \cdot \frac{0.75 + 0.125 \cdot \frac{1}{t}}{t} - 4.5\right)\right)}} \]

11. Taylor expanded in t around 0 12.4%

    \[\leadsto \color{blue}{0.8333333333333334 + -8 \cdot t} \]
12. Step-by-step derivation

    1. *-commutative 12.4%

       \[\leadsto 0.8333333333333334 + \color{blue}{t \cdot -8} \]

13. Simplified 12.4%

    \[\leadsto \color{blue}{0.8333333333333334 + t \cdot -8} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(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))))))))