Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 10.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{\frac{1}{t} - -1}\\ t_2 := \mathsf{fma}\left({t}^{-1}, t, t\right)\\ \frac{\left(1 + \frac{\frac{4}{t\_2} - 4}{t\_2}\right) + \left(4 + \frac{-4}{t\_2}\right)}{t\_1 \cdot t\_1 + 2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (- (/ 1.0 t) -1.0))))
        (t_2 (fma (pow t -1.0) t t)))
   (/
    (+ (+ 1.0 (/ (- (/ 4.0 t_2) 4.0) t_2)) (+ 4.0 (/ -4.0 t_2)))
    (+ (* t_1 t_1) 2.0))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / ((1.0 / t) - -1.0));
	double t_2 = fma(pow(t, -1.0), t, t);
	return ((1.0 + (((4.0 / t_2) - 4.0) / t_2)) + (4.0 + (-4.0 / t_2))) / ((t_1 * t_1) + 2.0);
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)))
	t_2 = fma((t ^ -1.0), t, t)
	return Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / t_2) - 4.0) / t_2)) + Float64(4.0 + Float64(-4.0 / t_2))) / Float64(Float64(t_1 * t_1) + 2.0))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[t, -1.0], $MachinePrecision] * t + t), $MachinePrecision]}, N[(N[(N[(1.0 + N[(N[(N[(4.0 / t$95$2), $MachinePrecision] - 4.0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(4.0 + N[(-4.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{\left(1 + \frac{\frac{4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} - 4}{\mathsf{fma}\left({t}^{-1}, t, t\right)}\right) + \left(4 + \frac{-4}{\mathsf{fma}\left({t}^{-1}, t, t\right)}\right)}{\left(2 - \frac{\frac{2}{t}}{\frac{1}{t} - -1}\right) \cdot \left(2 - \frac{\frac{2}{t}}{\frac{1}{t} - -1}\right) + 2} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / t) - (-1.0d0)))
    t_2 = t_1 * t_1
    code = (t_2 + 1.0d0) / (t_2 + 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{\frac{1}{t} - -1}\\ \frac{\left(\frac{\frac{4}{1 + t} - 4}{1 + t} + 5\right) + \frac{-4}{1 + t}}{t\_1 \cdot t\_1 + 2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)))))
   (/
    (+ (+ (/ (- (/ 4.0 (+ 1.0 t)) 4.0) (+ 1.0 t)) 5.0) (/ -4.0 (+ 1.0 t)))
    (+ (* t_1 t_1) 2.0))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / ((1.0 / t) - -1.0));
	return (((((4.0 / (1.0 + t)) - 4.0) / (1.0 + t)) + 5.0) + (-4.0 / (1.0 + t))) / ((t_1 * t_1) + 2.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / ((1.0d0 / t) - (-1.0d0)))
    code = (((((4.0d0 / (1.0d0 + t)) - 4.0d0) / (1.0d0 + t)) + 5.0d0) + ((-4.0d0) / (1.0d0 + t))) / ((t_1 * t_1) + 2.0d0)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / ((1.0 / t) - -1.0));
	return (((((4.0 / (1.0 + t)) - 4.0) / (1.0 + t)) + 5.0) + (-4.0 / (1.0 + t))) / ((t_1 * t_1) + 2.0);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / ((1.0 / t) - -1.0))
	return (((((4.0 / (1.0 + t)) - 4.0) / (1.0 + t)) + 5.0) + (-4.0 / (1.0 + t))) / ((t_1 * t_1) + 2.0)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)))
	return Float64(Float64(Float64(Float64(Float64(Float64(4.0 / Float64(1.0 + t)) - 4.0) / Float64(1.0 + t)) + 5.0) + Float64(-4.0 / Float64(1.0 + t))) / Float64(Float64(t_1 * t_1) + 2.0))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / ((1.0 / t) - -1.0));
	tmp = (((((4.0 / (1.0 + t)) - 4.0) / (1.0 + t)) + 5.0) + (-4.0 / (1.0 + t))) / ((t_1 * t_1) + 2.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 5.0), $MachinePrecision] + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-plus N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{-4}{{t}^{\color{blue}{0}} + t} + \left(4 + \left(\frac{\frac{4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} - 4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} + 1\right)\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-+r+ N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 99.8

       \[\leadsto \frac{\frac{-4}{1 + t} + \left(\color{blue}{5} + \frac{\frac{4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} - 4}{\mathsf{fma}\left({t}^{-1}, t, t\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{\left(\frac{\frac{4}{1 + t} - 4}{1 + t} + 5\right) + \frac{-4}{1 + t}}{\left(2 - \frac{\frac{2}{t}}{\frac{1}{t} - -1}\right) \cdot \left(2 - \frac{\frac{2}{t}}{\frac{1}{t} - -1}\right) + 2} \]
8. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{4}{1 + t} - 4}{1 + t} + 5\right) + \frac{-4}{1 + t}}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))
   (/
    (+ (+ (/ (- (/ 4.0 (+ 1.0 t)) 4.0) (+ 1.0 t)) 5.0) (/ -4.0 (+ 1.0 t)))
    (+ (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t) 2.0))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	} else {
		tmp = (((((4.0 / (1.0 + t)) - 4.0) / (1.0 + t)) + 5.0) + (-4.0 / (1.0 + t))) / (((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(4.0 / Float64(1.0 + t)) - 4.0) / Float64(1.0 + t)) + 5.0) + Float64(-4.0 / Float64(1.0 + t))) / Float64(Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 5.0), $MachinePrecision] + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

       2. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

       3. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

       5. mul-1-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}}{t} \]

       6. unsub-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9} - \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9} - \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

       9. +-commutative N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{4}{81} \cdot \frac{1}{t} + \frac{1}{27}}}{t}}{t} \]

       10. lower-+.f64 N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{4}{81} \cdot \frac{1}{t} + \frac{1}{27}}}{t}}{t} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{\frac{4}{81} \cdot 1}{t}} + \frac{1}{27}}{t}}{t} \]

       12. metadata-eval N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\frac{\color{blue}{\frac{4}{81}}}{t} + \frac{1}{27}}{t}}{t} \]

       13. lower-/.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{-4}{{t}^{\color{blue}{0}} + t} + \left(4 + \left(\frac{\frac{4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} - 4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} + 1\right)\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 99.6

          \[\leadsto \frac{\frac{-4}{1 + t} + \left(\color{blue}{5} + \frac{\frac{4}{\mathsf{fma}\left({t}^{-1}, t, t\right)} - 4}{\mathsf{fma}\left({t}^{-1}, t, t\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 100.0

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

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{-4}{1 + t} + \left(5 + \frac{\frac{4}{1 + t} - 4}{1 + t}\right)}{2 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{4}{1 + t} - 4}{1 + t} + 5\right) + \frac{-4}{1 + t}}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
     t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

       2. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

       3. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]

       5. mul-1-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}}{t} \]

       6. unsub-neg N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9} - \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9} - \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \color{blue}{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}}{t} \]

       9. +-commutative N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{4}{81} \cdot \frac{1}{t} + \frac{1}{27}}}{t}}{t} \]

       10. lower-+.f64 N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{4}{81} \cdot \frac{1}{t} + \frac{1}{27}}}{t}}{t} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{\frac{4}{81} \cdot 1}{t}} + \frac{1}{27}}{t}}{t} \]

       12. metadata-eval N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\frac{\color{blue}{\frac{4}{81}}}{t} + \frac{1}{27}}{t}}{t} \]

       13. lower-/.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

       7. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

       8. unpow2 N/A

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

       9. lower-*.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   (-
    0.8333333333333334
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

       2. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{27}}{{t}^{2}} + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

       3. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \frac{\frac{1}{27}}{{t}^{2}}} \]

       4. associate--r- N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

       6. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{\color{blue}{t \cdot t}}\right) \]

       8. associate-/r* N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \color{blue}{\frac{\frac{\frac{1}{27}}{t}}{t}}\right) \]

       9. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\color{blue}{\frac{1}{27} \cdot 1}}{t}}{t}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\color{blue}{\frac{1}{27} \cdot \frac{1}{t}}}{t}\right) \]

       11. div-sub N/A

           \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

       12. lower--.f64 N/A

           \[\leadsto \color{blue}{\frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

       14. lower--.f64 N/A

           \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}}{t} \]

       15. associate-*r/ N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \color{blue}{\frac{\frac{1}{27} \cdot 1}{t}}}{t} \]

       16. metadata-eval N/A

           \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{\color{blue}{\frac{1}{27}}}{t}}{t} \]

       17. lower-/.f64 99.8

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

   11. Applied rewrites 99.8%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

       7. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

       8. unpow2 N/A

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

       9. lower-*.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

       4. lower-/.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

       7. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

       8. unpow2 N/A

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

       9. lower-*.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (fma -2.0 t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

       4. lower-/.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

       6. unpow2 N/A

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

       7. lower-*.f64 99.8

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

   11. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(t, t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(t, t, 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(t * t + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

       4. lower-/.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

       3. lower-fma.f64 99.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 0.0005)
   0.8333333333333334
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) - -1.0)) <= 0.0005) {
		tmp = 0.8333333333333334;
	} else {
		tmp = fma(t, t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) - -1.0)) <= 0.0005)
		tmp = 0.8333333333333334;
	else
		tmp = fma(t, t, 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 0.0005], 0.8333333333333334, N[(t * t + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 5.0000000000000001e-4

   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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.1%

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

   if 5.0000000000000001e-4 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

       3. lower-fma.f64 99.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 0.0005:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (- (/ 1.0 t) -1.0)) 1.0) 0.8333333333333334 0.5))

C

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

Fortran

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

Java

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

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / ((1.0 / t) - -1.0)) <= 1.0:
		tmp = 0.8333333333333334
	else:
		tmp = 0.5
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], 1.0], 0.8333333333333334, 0.5]

Derivation

1. Split input into 2 regimes

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

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

   5. Applied rewrites 16.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{6}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \frac{1}{\color{blue}{6}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} - -1} \leq 1:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.1% accurate, 184.0× 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

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

5. Applied rewrites 55.9%

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

6. Taylor expanded in t around inf

   \[\leadsto \frac{1}{\color{blue}{6}} \]
7. Step-by-step derivation

8. Applied rewrites 17.2%

   \[\leadsto \frac{1}{\color{blue}{6}} \]

9. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
10. Step-by-step derivation

11. Applied rewrites 57.4%

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

Reproduce

herbie shell --seed 2024308 
(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))))))))