Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 17.4s

Alternatives: 7

Speedup: 1.5×

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: 99.9% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(-2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(t$95$1 * t$95$1), $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. div-inv 100.0%

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

   2. associate-/l* 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. +-commutative 100.0%

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

   9. distribute-lft-in 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. rgt-mult-inverse 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-rgt-identity 100.0%

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

   14. unsub-neg 100.0%

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

6. Simplified 100.0%

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

   1. div-inv 100.0%

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

   2. associate-/l* 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. +-commutative 100.0%

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

   9. distribute-lft-in 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. rgt-mult-inverse 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-rgt-identity 100.0%

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

   14. unsub-neg 100.0%

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

10. Simplified 100.0%

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

    1. div-inv 100.0%

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

    2. associate-/l* 100.0%

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

12. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. associate-*r/ 100.0%

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

    3. metadata-eval 100.0%

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

    4. metadata-eval 100.0%

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

    5. associate-/r* 100.0%

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

    6. neg-mul-1 100.0%

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

    7. distribute-lft-neg-in 100.0%

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

    8. +-commutative 100.0%

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

    9. distribute-lft-in 100.0%

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

    10. distribute-lft-neg-out 100.0%

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

    11. rgt-mult-inverse 100.0%

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

    12. metadata-eval 100.0%

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

    13. *-rgt-identity 100.0%

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

    14. unsub-neg 100.0%

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

14. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. metadata-eval 100.0%

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

16. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. metadata-eval 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. distribute-neg-frac2 100.0%

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

    5. distribute-lft-neg-in 100.0%

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

    6. +-commutative 100.0%

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

    7. distribute-lft-in 100.0%

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

    8. distribute-lft-neg-out 100.0%

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

    9. rgt-mult-inverse 100.0%

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

    10. metadata-eval 100.0%

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

    11. *-rgt-identity 100.0%

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

    12. unsub-neg 100.0%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.35 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(t \cdot -2 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.35) (not (<= t 0.58)))
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   (+ 0.5 (* (* t t) (+ (* t -2.0) 1.0)))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.35) || !(t <= 0.58)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.35d0)) .or. (.not. (t <= 0.58d0))) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = 0.5d0 + ((t * t) * ((t * (-2.0d0)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.35) || !(t <= 0.58)) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.35) or not (t <= 0.58):
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.35) || !(t <= 0.58))
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(Float64(t * -2.0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.35) || ~((t <= 0.58)))
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.35], N[Not[LessEqual[t, 0.58]], $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[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(N[(t * -2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.34999999999999998 or 0.57999999999999996 < 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 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. mul-1-neg 99.0%

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

      4. unsub-neg 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

   5. Simplified 99.0%

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

   if -0.34999999999999998 < t < 0.57999999999999996

   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 99.5%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. unpow2 99.5%

         \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + -2 \cdot t\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + -2 \cdot t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(t \cdot -2 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.45\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(t \cdot -2 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.6) (not (<= t 0.45)))
   (+
    0.8333333333333334
    (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))
   (+ 0.5 (* (* t t) (+ (* t -2.0) 1.0)))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.6) || !(t <= 0.45)) {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.6d0)) .or. (.not. (t <= 0.45d0))) then
        tmp = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    else
        tmp = 0.5d0 + ((t * t) * ((t * (-2.0d0)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.6) || !(t <= 0.45)) {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.6) or not (t <= 0.45):
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	else:
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.6) || !(t <= 0.45))
		tmp = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t));
	else
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(Float64(t * -2.0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.6) || ~((t <= 0.45)))
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	else
		tmp = 0.5 + ((t * t) * ((t * -2.0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.6], N[Not[LessEqual[t, 0.45]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(N[(t * -2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.450000000000000011 < 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 98.8%

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

      1. associate-*r/ 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-lft-in 98.8%

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

      4. metadata-eval 98.8%

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

      5. mul-1-neg 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. distribute-rgt-neg-out 98.8%

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

      8. remove-double-neg 98.8%

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

      9. associate-*r/ 98.8%

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

      10. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   if -0.599999999999999978 < t < 0.450000000000000011

   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 99.5%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. unpow2 99.5%

         \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + -2 \cdot t\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto 0.5 + \color{blue}{\left(t \cdot t\right)} \cdot \left(1 + -2 \cdot t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.45\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(t \cdot -2 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if ((t <= -0.5) || !(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 <= (-0.5d0)) .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 <= -0.5) || !(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 <= -0.5) 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 <= -0.5) || !(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 <= -0.5) || ~((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, -0.5], 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 < -0.5 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 98.8%

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

      1. associate-*r/ 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-lft-in 98.8%

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

      4. metadata-eval 98.8%

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

      5. mul-1-neg 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. distribute-rgt-neg-out 98.8%

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

      8. remove-double-neg 98.8%

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

      9. associate-*r/ 98.8%

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

      10. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   if -0.5 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.2%

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

4. Final simplification 98.5%

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

Alternative 5: 98.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		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 <= 0.66d0))) 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 <= 0.66)) {
		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 <= 0.66):
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		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 <= 0.66)))
		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, 0.66]], $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 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.6%

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

      1. sub-neg 98.6%

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

      2. associate-*r/ 98.6%

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

      3. metadata-eval 98.6%

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

      4. distribute-neg-frac 98.6%

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

      5. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% 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 98.0%

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

   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 98.2%

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

Alternative 7: 59.3% 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 56.5%

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

Reproduce

herbie shell --seed 2024096 
(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))))))))