Kahan p13 Example 2

Percentage Accurate: 100.0% → 99.9%

Time: 9.9s

Alternatives: 9

Speedup: N/A×

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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{1}{t}\\ \frac{1 + \left(2 \cdot \left(2 + \frac{\frac{-2}{t}}{t\_1}\right) + \frac{\frac{\frac{\frac{-4}{t}}{t\_1} - -4}{t}}{-1 - \frac{1}{t}}\right)}{2 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(2 + \frac{-2}{1 + t}\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ 1.0 t))))
   (/
    (+
     1.0
     (+
      (* 2.0 (+ 2.0 (/ (/ -2.0 t) t_1)))
      (/ (/ (- (/ (/ -4.0 t) t_1) -4.0) t) (- -1.0 (/ 1.0 t)))))
    (+ 2.0 (* (+ 2.0 (/ 2.0 (- -1.0 t))) (+ 2.0 (/ -2.0 (+ 1.0 t))))))))

C

double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	return (1.0 + ((2.0 * (2.0 + ((-2.0 / t) / t_1))) + (((((-4.0 / t) / t_1) - -4.0) / t) / (-1.0 - (1.0 / t))))) / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 + (-2.0 / (1.0 + t)))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 1.0d0 + (1.0d0 / t)
    code = (1.0d0 + ((2.0d0 * (2.0d0 + (((-2.0d0) / t) / t_1))) + ((((((-4.0d0) / t) / t_1) - (-4.0d0)) / t) / ((-1.0d0) - (1.0d0 / t))))) / (2.0d0 + ((2.0d0 + (2.0d0 / ((-1.0d0) - t))) * (2.0d0 + ((-2.0d0) / (1.0d0 + t)))))
end function

Java

public static double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	return (1.0 + ((2.0 * (2.0 + ((-2.0 / t) / t_1))) + (((((-4.0 / t) / t_1) - -4.0) / t) / (-1.0 - (1.0 / t))))) / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 + (-2.0 / (1.0 + t)))));
}

Python

def code(t):
	t_1 = 1.0 + (1.0 / t)
	return (1.0 + ((2.0 * (2.0 + ((-2.0 / t) / t_1))) + (((((-4.0 / t) / t_1) - -4.0) / t) / (-1.0 - (1.0 / t))))) / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 + (-2.0 / (1.0 + t)))))

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(1.0 / t))
	return Float64(Float64(1.0 + Float64(Float64(2.0 * Float64(2.0 + Float64(Float64(-2.0 / t) / t_1))) + Float64(Float64(Float64(Float64(Float64(-4.0 / t) / t_1) - -4.0) / t) / Float64(-1.0 - Float64(1.0 / t))))) / Float64(2.0 + Float64(Float64(2.0 + Float64(2.0 / Float64(-1.0 - t))) * Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))))))
end

MATLAB

function tmp = code(t)
	t_1 = 1.0 + (1.0 / t);
	tmp = (1.0 + ((2.0 * (2.0 + ((-2.0 / t) / t_1))) + (((((-4.0 / t) / t_1) - -4.0) / t) / (-1.0 - (1.0 / t))))) / (2.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 + (-2.0 / (1.0 + t)))));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(N[(2.0 * N[(2.0 + N[(N[(-2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(-4.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision] - -4.0), $MachinePrecision] / t), $MachinePrecision] / N[(-1.0 - N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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. sub-neg 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-neg-frac 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. sub-neg 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. distribute-neg-frac 100.0%

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

4. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

8. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

10. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

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

    2. associate-/r* 100.0%

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

    3. distribute-lft-in 100.0%

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

    4. *-rgt-identity 100.0%

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

    5. rgt-mult-inverse 100.0%

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

12. Simplified 100.0%

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

    1. associate-*r/ 100.0%

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

14. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. distribute-rgt-neg-in 100.0%

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

    4. *-commutative 100.0%

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

    5. distribute-lft-in 100.0%

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

    6. metadata-eval 100.0%

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

    7. distribute-neg-in 100.0%

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

    8. metadata-eval 100.0%

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

    9. unsub-neg 100.0%

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

    10. associate-*r/ 100.0%

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

    11. associate-*r/ 100.0%

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

    12. metadata-eval 100.0%

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

16. Simplified 100.0%

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

17. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 + ((2.0 / t) / (-1.0 - (1.0 / t)));
	double t_2 = 2.0 + (2.0 / (-1.0 - t));
	return (1.0 + (t_1 * t_1)) / (2.0 + (t_2 * 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 = 2.0d0 + (2.0d0 / ((-1.0d0) - t))
    code = (1.0d0 + (t_1 * t_1)) / (2.0d0 + (t_2 * 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 = 2.0 + (2.0 / (-1.0 - t));
	return (1.0 + (t_1 * t_1)) / (2.0 + (t_2 * t_2));
}

Python

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 + ((2.0 / t) / (-1.0 - (1.0 / t)));
	t_2 = 2.0 + (2.0 / (-1.0 - t));
	tmp = (1.0 + (t_1 * t_1)) / (2.0 + (t_2 * 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[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$2 * t$95$2), $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. *-un-lft-identity 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/r* 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. *-rgt-identity 100.0%

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

   5. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/r* 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. *-rgt-identity 100.0%

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

   5. rgt-mult-inverse 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.74\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(2 \cdot t\right)}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.74)))
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   (/
    (+ 1.0 (* (+ 2.0 (/ 2.0 (- -1.0 t))) (* 2.0 t)))
    (+ 2.0 (* (* 2.0 t) (* 2.0 t))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.74)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 * t))) / (2.0 + ((2.0 * t) * (2.0 * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.74d0))) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = (1.0d0 + ((2.0d0 + (2.0d0 / ((-1.0d0) - t))) * (2.0d0 * t))) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.74)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 * t))) / (2.0 + ((2.0 * t) * (2.0 * t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.74):
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 * t))) / (2.0 + ((2.0 * t) * (2.0 * t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.74))
		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(2.0 / Float64(-1.0 - t))) * Float64(2.0 * t))) / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.74)))
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = (1.0 + ((2.0 + (2.0 / (-1.0 - t))) * (2.0 * t))) / (2.0 + ((2.0 * t) * (2.0 * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.74]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.73999999999999999 < 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 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   if -0.52000000000000002 < t < 0.73999999999999999

   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 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in t around 0 99.2%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.2%

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

       1. *-un-lft-identity 100.0%

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

   13. Applied egg-rr 99.2%

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

       1. *-lft-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. distribute-lft-in 100.0%

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

       4. *-rgt-identity 100.0%

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

       5. rgt-mult-inverse 100.0%

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

   15. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.74\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 + \frac{2}{-1 - t}\right) \cdot \left(2 \cdot t\right)}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)\\ \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_1}{2 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* (* 2.0 t) (* 2.0 t))))
   (if (or (<= t -0.52) (not (<= t 0.7)))
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t))
     (/ (+ 1.0 t_1) (+ 2.0 t_1)))))

C

double code(double t) {
	double t_1 = (2.0 * t) * (2.0 * t);
	double tmp;
	if ((t <= -0.52) || !(t <= 0.7)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	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) * (2.0d0 * t)
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.7d0))) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) * (2.0 * t);
	double tmp;
	if ((t <= -0.52) || !(t <= 0.7)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = (2.0 * t) * (2.0 * t)
	tmp = 0
	if (t <= -0.52) or not (t <= 0.7):
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) * Float64(2.0 * t))
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.7))
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (2.0 * t) * (2.0 * t);
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.7)))
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.7]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.69999999999999996 < 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 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   if -0.52000000000000002 < t < 0.69999999999999996

   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 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in t around 0 99.2%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in t around 0 99.1%

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

       1. *-commutative 99.1%

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

   14. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.7\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.33d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   if -0.330000000000000016 < 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 99.0%

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

4. Final simplification 99.3%

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

Alternative 6: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-neg-frac 99.4%

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

      7. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

   if -0.52000000000000002 < 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 99.0%

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

4. Final simplification 99.2%

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

Alternative 7: 98.9% 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 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   5. Simplified 98.9%

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

   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 99.0%

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

4. Final simplification 98.9%

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

Alternative 8: 98.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.1%

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

   if -0.330000000000000016 < 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 99.0%

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

Alternative 9: 59.4% 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 59.3%

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

Reproduce

herbie shell --seed 2024145 
(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))))))))