Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 16.5s

Alternatives: 10

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $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. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

4. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. metadata-eval 100.0%

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

   3. +-commutative 100.0%

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

   4. log1p-undefine 100.0%

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

   5. rem-exp-log 100.0%

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

   6. associate-+r+ 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-lft-identity 100.0%

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

   9. associate-/r* 100.0%

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

   10. distribute-lft-in 100.0%

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

   11. rgt-mult-inverse 100.0%

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

   12. *-rgt-identity 100.0%

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

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. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

8. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. metadata-eval 100.0%

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

   3. +-commutative 100.0%

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

   4. log1p-undefine 100.0%

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

   5. rem-exp-log 100.0%

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

   6. associate-+r+ 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-lft-identity 100.0%

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

   9. associate-/r* 100.0%

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

   10. distribute-lft-in 100.0%

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

   11. rgt-mult-inverse 100.0%

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

   12. *-rgt-identity 100.0%

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

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. Step-by-step derivation

    1. expm1-log1p-u 100.0%

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

    2. expm1-undefine 100.0%

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

12. Applied egg-rr 100.0%

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

    1. sub-neg 100.0%

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

    2. metadata-eval 100.0%

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

    3. +-commutative 100.0%

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

    4. log1p-undefine 100.0%

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

    5. rem-exp-log 100.0%

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

    6. associate-+r+ 100.0%

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

    7. metadata-eval 100.0%

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

    8. +-lft-identity 100.0%

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

    9. associate-/r* 100.0%

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

    10. distribute-lft-in 100.0%

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

    11. rgt-mult-inverse 100.0%

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

    12. *-rgt-identity 100.0%

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

14. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. metadata-eval 100.0%

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

16. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. distribute-lft-in 100.0%

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

    3. rgt-mult-inverse 100.0%

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

    4. *-rgt-identity 100.0%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ t_2 := 2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\\ \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + -2 \cdot t\right)\\ \mathbf{elif}\;t \leq 115000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (-
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            0.2222222222222222)
           t)))
        (t_2 (+ 2.0 (/ (/ 2.0 t) (+ -1.0 (/ -1.0 t))))))
   (if (<= t -0.35)
     t_1
     (if (<= t 0.58)
       (+ 0.5 (* (* t t) (+ 1.0 (* -2.0 t))))
       (if (<= t 115000000.0)
         t_1
         (/ (+ 1.0 (- 4.0 (/ 8.0 t))) (+ 2.0 (* t_2 t_2))))))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	double t_2 = 2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)));
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	} else if (t <= 115000000.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (4.0 - (8.0 / t))) / (2.0 + (t_2 * t_2));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    t_2 = 2.0d0 + ((2.0d0 / t) / ((-1.0d0) + ((-1.0d0) / t)))
    if (t <= (-0.35d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + ((-2.0d0) * t)))
    else if (t <= 115000000.0d0) then
        tmp = t_1
    else
        tmp = (1.0d0 + (4.0d0 - (8.0d0 / t))) / (2.0d0 + (t_2 * t_2))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	double t_2 = 2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)));
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	} else if (t <= 115000000.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (4.0 - (8.0 / t))) / (2.0 + (t_2 * t_2));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	t_2 = 2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)))
	tmp = 0
	if t <= -0.35:
		tmp = t_1
	elif t <= 0.58:
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)))
	elif t <= 115000000.0:
		tmp = t_1
	else:
		tmp = (1.0 + (4.0 - (8.0 / t))) / (2.0 + (t_2 * t_2))
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / t) / Float64(-1.0 + Float64(-1.0 / t))))
	tmp = 0.0
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(-2.0 * t))));
	elseif (t <= 115000000.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(4.0 - Float64(8.0 / t))) / Float64(2.0 + Float64(t_2 * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	t_2 = 2.0 + ((2.0 / t) / (-1.0 + (-1.0 / t)));
	tmp = 0.0;
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	elseif (t <= 115000000.0)
		tmp = t_1;
	else
		tmp = (1.0 + (4.0 - (8.0 / t))) / (2.0 + (t_2 * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.35], t$95$1, If[LessEqual[t, 0.58], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 115000000.0], t$95$1, N[(N[(1.0 + N[(4.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.34999999999999998 or 0.57999999999999996 < t < 1.15e8

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. mul-1-neg 97.9%

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

      4. unsub-neg 97.9%

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

      5. associate-*r/ 97.9%

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

      6. metadata-eval 97.9%

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

   5. Simplified 97.9%

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

   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. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. unpow2 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1.15e8 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + -2 \cdot t\right)\\ \mathbf{elif}\;t \leq 115000000:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(4 - \frac{8}{t}\right)}{2 + \left(2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\right) \cdot \left(2 + \frac{\frac{2}{t}}{-1 + \frac{-1}{t}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + -2 \cdot t\right)\\ \mathbf{elif}\;t \leq 2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (-
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            0.2222222222222222)
           t))))
   (if (<= t -0.35)
     t_1
     (if (<= t 0.58)
       (+ 0.5 (* (* t t) (+ 1.0 (* -2.0 t))))
       (if (<= t 2000000000.0)
         t_1
         (if (<= t 1.2e+16)
           (+ (- 1.8333333333333333 (/ 0.2222222222222222 t)) -1.0)
           0.8333333333333334))))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    if (t <= (-0.35d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + ((-2.0d0) * t)))
    else if (t <= 2000000000.0d0) then
        tmp = t_1
    else if (t <= 1.2d+16) then
        tmp = (1.8333333333333333d0 - (0.2222222222222222d0 / t)) + (-1.0d0)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	tmp = 0
	if t <= -0.35:
		tmp = t_1
	elif t <= 0.58:
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)))
	elif t <= 2000000000.0:
		tmp = t_1
	elif t <= 1.2e+16:
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(-2.0 * t))));
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = Float64(Float64(1.8333333333333333 - Float64(0.2222222222222222 / t)) + -1.0);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.35], t$95$1, If[LessEqual[t, 0.58], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2000000000.0], t$95$1, If[LessEqual[t, 1.2e+16], N[(N[(1.8333333333333333 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 0.8333333333333334]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.34999999999999998 or 0.57999999999999996 < t < 2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. mul-1-neg 97.9%

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

      4. unsub-neg 97.9%

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

      5. associate-*r/ 97.9%

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

      6. metadata-eval 97.9%

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

   5. Simplified 97.9%

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

   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. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. unpow2 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 2e9 < t < 1.2e16

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-undefine 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} - 1} \]

      2. sub-neg 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} + \left(-1\right)} \]

      3. log1p-undefine 98.8%

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

      4. rem-exp-log 98.8%

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

      5. associate-+r- 97.5%

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

      6. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{1.8333333333333333} - \frac{0.2222222222222222}{t}\right) + \left(-1\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + \color{blue}{-1} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1} \]

   if 1.2e16 < 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 100.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + -2 \cdot t\right)\\ \mathbf{elif}\;t \leq 2000000000:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + -2 \cdot t\right)\\ \mathbf{elif}\;t \leq 2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (-
          0.8333333333333334
          (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))
   (if (<= t -0.6)
     t_1
     (if (<= t 0.45)
       (+ 0.5 (* (* t t) (+ 1.0 (* -2.0 t))))
       (if (<= t 2000000000.0)
         t_1
         (if (<= t 1.2e+16)
           (+ (- 1.8333333333333333 (/ 0.2222222222222222 t)) -1.0)
           0.8333333333333334))))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1;
	} else if (t <= 0.45) {
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    if (t <= (-0.6d0)) then
        tmp = t_1
    else if (t <= 0.45d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + ((-2.0d0) * t)))
    else if (t <= 2000000000.0d0) then
        tmp = t_1
    else if (t <= 1.2d+16) then
        tmp = (1.8333333333333333d0 - (0.2222222222222222d0 / t)) + (-1.0d0)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1;
	} else if (t <= 0.45) {
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.6:
		tmp = t_1
	elif t <= 0.45:
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)))
	elif t <= 2000000000.0:
		tmp = t_1
	elif t <= 1.2e+16:
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.6)
		tmp = t_1;
	elseif (t <= 0.45)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(-2.0 * t))));
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = Float64(Float64(1.8333333333333333 - Float64(0.2222222222222222 / t)) + -1.0);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.6)
		tmp = t_1;
	elseif (t <= 0.45)
		tmp = 0.5 + ((t * t) * (1.0 + (-2.0 * t)));
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.6], t$95$1, If[LessEqual[t, 0.45], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2000000000.0], t$95$1, If[LessEqual[t, 1.2e+16], N[(N[(1.8333333333333333 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 0.8333333333333334]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.599999999999999978 or 0.450000000000000011 < t < 2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. associate-*r/ 97.6%

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

      5. metadata-eval 97.6%

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

      6. distribute-neg-frac 97.6%

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

      7. metadata-eval 97.6%

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

   5. Simplified 97.6%

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

   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. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. unpow2 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 2e9 < t < 1.2e16

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-undefine 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} - 1} \]

      2. sub-neg 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} + \left(-1\right)} \]

      3. log1p-undefine 98.8%

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

      4. rem-exp-log 98.8%

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

      5. associate-+r- 97.5%

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

      6. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{1.8333333333333333} - \frac{0.2222222222222222}{t}\right) + \left(-1\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + \color{blue}{-1} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1} \]

   if 1.2e16 < 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 100.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + -2 \cdot t\right)\\ \mathbf{elif}\;t \leq 2000000000:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t \leq 2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (-
          0.8333333333333334
          (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))))
   (if (<= t -0.5)
     t_1
     (if (<= t 0.23)
       0.5
       (if (<= t 2000000000.0)
         t_1
         (if (<= t 1.2e+16)
           (+ (- 1.8333333333333333 (/ 0.2222222222222222 t)) -1.0)
           0.8333333333333334))))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.5) {
		tmp = t_1;
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    if (t <= (-0.5d0)) then
        tmp = t_1
    else if (t <= 0.23d0) then
        tmp = 0.5d0
    else if (t <= 2000000000.0d0) then
        tmp = t_1
    else if (t <= 1.2d+16) then
        tmp = (1.8333333333333333d0 - (0.2222222222222222d0 / t)) + (-1.0d0)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.5) {
		tmp = t_1;
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.5:
		tmp = t_1
	elif t <= 0.23:
		tmp = 0.5
	elif t <= 2000000000.0:
		tmp = t_1
	elif t <= 1.2e+16:
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.5)
		tmp = t_1;
	elseif (t <= 0.23)
		tmp = 0.5;
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = Float64(Float64(1.8333333333333333 - Float64(0.2222222222222222 / t)) + -1.0);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.5)
		tmp = t_1;
	elseif (t <= 0.23)
		tmp = 0.5;
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.5], t$95$1, If[LessEqual[t, 0.23], 0.5, If[LessEqual[t, 2000000000.0], t$95$1, If[LessEqual[t, 1.2e+16], N[(N[(1.8333333333333333 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 0.8333333333333334]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.5 or 0.23000000000000001 < t < 2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. associate-*r/ 97.6%

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

      5. metadata-eval 97.6%

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

      6. distribute-neg-frac 97.6%

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

      7. metadata-eval 97.6%

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

   5. Simplified 97.6%

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

   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%

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

   if 2e9 < t < 1.2e16

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-undefine 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} - 1} \]

      2. sub-neg 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} + \left(-1\right)} \]

      3. log1p-undefine 98.8%

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

      4. rem-exp-log 98.8%

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

      5. associate-+r- 97.5%

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

      6. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{1.8333333333333333} - \frac{0.2222222222222222}{t}\right) + \left(-1\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + \color{blue}{-1} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1} \]

   if 1.2e16 < 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 100.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t \leq 2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 0.8333333333333334 (/ 0.2222222222222222 t))))
   (if (<= t -0.49)
     t_1
     (if (<= t 0.66)
       0.5
       (if (<= t 2000000000.0)
         t_1
         (if (<= t 1.2e+16)
           (+ (- 1.8333333333333333 (/ 0.2222222222222222 t)) -1.0)
           0.8333333333333334))))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - (0.2222222222222222 / t);
	double tmp;
	if (t <= -0.49) {
		tmp = t_1;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    if (t <= (-0.49d0)) then
        tmp = t_1
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else if (t <= 2000000000.0d0) then
        tmp = t_1
    else if (t <= 1.2d+16) then
        tmp = (1.8333333333333333d0 - (0.2222222222222222d0 / t)) + (-1.0d0)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - (0.2222222222222222 / t);
	double tmp;
	if (t <= -0.49) {
		tmp = t_1;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else if (t <= 2000000000.0) {
		tmp = t_1;
	} else if (t <= 1.2e+16) {
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - (0.2222222222222222 / t)
	tmp = 0
	if t <= -0.49:
		tmp = t_1
	elif t <= 0.66:
		tmp = 0.5
	elif t <= 2000000000.0:
		tmp = t_1
	elif t <= 1.2e+16:
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.49)
		tmp = t_1;
	elseif (t <= 0.66)
		tmp = 0.5;
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = Float64(Float64(1.8333333333333333 - Float64(0.2222222222222222 / t)) + -1.0);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - (0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.49)
		tmp = t_1;
	elseif (t <= 0.66)
		tmp = 0.5;
	elseif (t <= 2000000000.0)
		tmp = t_1;
	elseif (t <= 1.2e+16)
		tmp = (1.8333333333333333 - (0.2222222222222222 / t)) + -1.0;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.49], t$95$1, If[LessEqual[t, 0.66], 0.5, If[LessEqual[t, 2000000000.0], t$95$1, If[LessEqual[t, 1.2e+16], N[(N[(1.8333333333333333 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 0.8333333333333334]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < t < 2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

   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%

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

   if 2e9 < t < 1.2e16

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-undefine 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} - 1} \]

      2. sub-neg 98.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} + \left(-1\right)} \]

      3. log1p-undefine 98.8%

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

      4. rem-exp-log 98.8%

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

      5. associate-+r- 97.5%

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

      6. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{1.8333333333333333} - \frac{0.2222222222222222}{t}\right) + \left(-1\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + \color{blue}{-1} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1} \]

   if 1.2e16 < 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 100.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 0.8333333333333334 (/ 0.2222222222222222 t))))
   (if (<= t -0.49)
     t_1
     (if (<= t 0.66) 0.5 (if (<= t 5e+14) t_1 0.8333333333333334)))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - (0.2222222222222222 / t);
	double tmp;
	if (t <= -0.49) {
		tmp = t_1;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else if (t <= 5e+14) {
		tmp = t_1;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    if (t <= (-0.49d0)) then
        tmp = t_1
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else if (t <= 5d+14) then
        tmp = t_1
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - (0.2222222222222222 / t);
	double tmp;
	if (t <= -0.49) {
		tmp = t_1;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else if (t <= 5e+14) {
		tmp = t_1;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - (0.2222222222222222 / t)
	tmp = 0
	if t <= -0.49:
		tmp = t_1
	elif t <= 0.66:
		tmp = 0.5
	elif t <= 5e+14:
		tmp = t_1
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.49)
		tmp = t_1;
	elseif (t <= 0.66)
		tmp = 0.5;
	elseif (t <= 5e+14)
		tmp = t_1;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - (0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.49)
		tmp = t_1;
	elseif (t <= 0.66)
		tmp = 0.5;
	elseif (t <= 5e+14)
		tmp = t_1;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.49], t$95$1, If[LessEqual[t, 0.66], 0.5, If[LessEqual[t, 5e+14], t$95$1, 0.8333333333333334]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < t < 5e14

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

   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%

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

   if 5e14 < t < 1.2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

      1. expm1-log1p-u 97.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]

   7. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-undefine 97.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} - 1} \]

      2. sub-neg 97.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} + \left(-1\right)} \]

      3. log1p-undefine 97.5%

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

      4. rem-exp-log 97.5%

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

      5. associate-+r- 97.5%

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

      6. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{1.8333333333333333} - \frac{0.2222222222222222}{t}\right) + \left(-1\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + \color{blue}{-1} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1} \]

   10. Taylor expanded in t around inf 100.0%

       \[\leadsto \color{blue}{1.8333333333333333} + -1 \]

   if 1.2e16 < 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 100.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+14}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t \leq 5:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 1.0) 0.5 (if (<= t 5.0) 0.8333333333333334 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 if (t <= 5.0) {
		tmp = 0.8333333333333334;
	} 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 if (t <= 5.0d0) then
        tmp = 0.8333333333333334d0
    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 if (t <= 5.0) {
		tmp = 0.8333333333333334;
	} 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
	elif t <= 5.0:
		tmp = 0.8333333333333334
	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;
	elseif (t <= 5.0)
		tmp = 0.8333333333333334;
	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;
	elseif (t <= 5.0)
		tmp = 0.8333333333333334;
	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, If[LessEqual[t, 5.0], 0.8333333333333334, 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. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t \leq 5:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 3 regimes

2. if t < -0.330000000000000016 or 1 < t < 5 or 1.2e16 < 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%

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

   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%

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

   if 5 < t < 1.2e16

   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 84.9%

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

      1. associate-*r/ 84.9%

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

      2. metadata-eval 84.9%

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

   5. Simplified 84.9%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]

   7. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-undefine 84.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} - 1} \]

      2. sub-neg 84.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} + \left(-1\right)} \]

      3. log1p-undefine 84.3%

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

      4. rem-exp-log 84.3%

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

      5. associate-+r- 83.7%

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

      6. metadata-eval 84.9%

         \[\leadsto \left(\color{blue}{1.8333333333333333} - \frac{0.2222222222222222}{t}\right) + \left(-1\right) \]

      7. metadata-eval 84.9%

         \[\leadsto \left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + \color{blue}{-1} \]

   9. Simplified 84.9%

      \[\leadsto \color{blue}{\left(1.8333333333333333 - \frac{0.2222222222222222}{t}\right) + -1} \]

   10. Taylor expanded in t around inf 71.5%

       \[\leadsto \color{blue}{1.8333333333333333} + -1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

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

Alternative 10: 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))))))))