Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 15.4s

Alternatives: 12

Speedup: 2.0×

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, 1.6× speedup

Math

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

FPCore

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

C

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

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(Float64(2.0 / t) / Float64(-1.0 - Float64(1.0 / t)))) * Float64(2.0 + Float64(2.0 / Float64(-1.0 - t))))) / fma(t_1, t_1, 2.0))
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[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 - N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1 + 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 100.0%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. sub-neg N/A

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

   3. +-lowering-+.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. /-lowering-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. rgt-mult-inverse N/A

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

   8. +-commutative N/A

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

   9. +-lowering-+.f64 N/A

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

   10. sub-neg N/A

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

   11. +-lowering-+.f64 N/A

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

   12. distribute-neg-frac N/A

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

   13. /-lowering-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. rgt-mult-inverse N/A

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 100.0

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

6. Applied egg-rr 100.0%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. /-lowering-/.f64 100.0

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

8. Applied egg-rr 100.0%

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. rgt-mult-inverse N/A

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

   4. +-lowering-+.f64 100.0

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 1\right)}{\mathsf{fma}\left(t \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(-2, t, 2\right), -2\right), 2\right), 2 \cdot t, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   (/
    (fma (* t t) (fma t (fma t 12.0 -8.0) 4.0) 1.0)
    (fma (* t (fma t (fma t (fma -2.0 t 2.0) -2.0) 2.0)) (* 2.0 t) 2.0))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = fma((t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 1.0) / fma((t * fma(t, fma(t, fma(-2.0, t, 2.0), -2.0), 2.0)), (2.0 * t), 2.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.001)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = Float64(fma(Float64(t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 1.0) / fma(Float64(t * fma(t, fma(t, fma(-2.0, t, 2.0), -2.0), 2.0)), Float64(2.0 * t), 2.0));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 12.0 + -8.0), $MachinePrecision] + 4.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(t * N[(t * N[(t * N[(-2.0 * t + 2.0), $MachinePrecision] + -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * 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))) < 1e-3

   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

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12}{{t}^{2}} - \color{blue}{\left(16 \cdot \frac{1}{{t}^{3}} + 8 \cdot \frac{1}{t}\right)}\right)} \]

      4. associate--r+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{{t}^{3}}\right) - 8 \cdot \frac{1}{t}\right)}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      7. associate-/r* N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \color{blue}{\frac{\frac{1}{t}}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - \color{blue}{\frac{16 \cdot \frac{1}{t}}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      9. div-sub N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{12 - 16 \cdot \frac{1}{t}}{{t}^{2}}} - 8 \cdot \frac{1}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12 - 16 \cdot \frac{1}{t}}{\color{blue}{t \cdot t}} - 8 \cdot \frac{1}{t}\right)} \]

      11. associate-/l/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}} - 8 \cdot \frac{1}{t}\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t} - \frac{\color{blue}{8}}{t}\right)} \]

      14. div-sub N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t} - 8}{t}}} \]

   5. Simplified 99.8%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \frac{-8 + \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   6. Taylor expanded in t around -inf

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

      1. +-lowering-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. /-lowering-/.f64 N/A

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

   8. Simplified 99.9%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}, \color{blue}{2 \cdot t}, 2\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 99.5

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

   10. Simplified 99.5%

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

   11. Taylor expanded in t around 0

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. accelerator-lowering-fma.f64 99.5

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

   13. Simplified 99.5%

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

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 1\right)}{\mathsf{fma}\left(4 + \frac{-4}{1 + t}, t, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   (/
    (fma (* t t) (fma t (fma t 12.0 -8.0) 4.0) 1.0)
    (fma (+ 4.0 (/ -4.0 (+ 1.0 t))) t 2.0))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = fma((t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 1.0) / fma((4.0 + (-4.0 / (1.0 + t))), t, 2.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.001)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = Float64(fma(Float64(t * t), fma(t, fma(t, 12.0, -8.0), 4.0), 1.0) / fma(Float64(4.0 + Float64(-4.0 / Float64(1.0 + t))), t, 2.0));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 12.0 + -8.0), $MachinePrecision] + 4.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(4.0 + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t + 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))) < 1e-3

   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

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12}{{t}^{2}} - \color{blue}{\left(16 \cdot \frac{1}{{t}^{3}} + 8 \cdot \frac{1}{t}\right)}\right)} \]

      4. associate--r+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{{t}^{3}}\right) - 8 \cdot \frac{1}{t}\right)}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      7. associate-/r* N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \color{blue}{\frac{\frac{1}{t}}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - \color{blue}{\frac{16 \cdot \frac{1}{t}}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      9. div-sub N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{12 - 16 \cdot \frac{1}{t}}{{t}^{2}}} - 8 \cdot \frac{1}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12 - 16 \cdot \frac{1}{t}}{\color{blue}{t \cdot t}} - 8 \cdot \frac{1}{t}\right)} \]

      11. associate-/l/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}} - 8 \cdot \frac{1}{t}\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t} - \frac{\color{blue}{8}}{t}\right)} \]

      14. div-sub N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t} - 8}{t}}} \]

   5. Simplified 99.8%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \frac{-8 + \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   6. Taylor expanded in t around -inf

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

      1. +-lowering-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. /-lowering-/.f64 N/A

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

   8. Simplified 99.9%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}, \color{blue}{2 \cdot t}, 2\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 99.5

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

   10. Simplified 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(t, \frac{1}{t}, t\right)}, \color{blue}{2 \cdot t}, 2\right)} \]
   11. Step-by-step derivation

       1. div-inv N/A

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

       2. rgt-mult-inverse N/A

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

       3. +-commutative N/A

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

       4. cancel-sign-sub-inv N/A

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

       5. metadata-eval N/A

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

       6. div-inv N/A

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

       7. associate-*r* N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-in N/A

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

       11. metadata-eval N/A

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

       12. +-lowering-+.f64 N/A

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

       13. associate-*r/ N/A

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

       14. /-lowering-/.f64 N/A

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

       15. metadata-eval N/A

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

       16. +-lowering-+.f64 99.5

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

   12. Applied egg-rr 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, \mathsf{fma}\left(t, 12, -8\right), 4\right), 1\right)}{\mathsf{fma}\left(4 + \frac{-4}{1 + t}, t, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 100.0%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. sub-neg N/A

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

   3. +-lowering-+.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. /-lowering-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. rgt-mult-inverse N/A

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

   8. +-commutative N/A

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

   9. +-lowering-+.f64 N/A

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

   10. sub-neg N/A

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

   11. +-lowering-+.f64 N/A

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

   12. distribute-neg-frac N/A

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

   13. /-lowering-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. rgt-mult-inverse N/A

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 100.0

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

6. Applied egg-rr 100.0%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. /-lowering-/.f64 100.0

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

8. Applied egg-rr 100.0%

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

   1. +-commutative N/A

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 5: 99.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (+
    0.8333333333333334
    (/
     (+
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      -0.2222222222222222)
     t))
   (fma (* t t) (fma t (+ t -2.0) 1.0) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	} else {
		tmp = fma((t * t), fma(t, (t + -2.0), 1.0), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.001)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t));
	else
		tmp = fma(Float64(t * t), fma(t, Float64(t + -2.0), 1.0), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] * N[(t * N[(t + -2.0), $MachinePrecision] + 1.0), $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))) < 1e-3

   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

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12}{{t}^{2}} - \color{blue}{\left(16 \cdot \frac{1}{{t}^{3}} + 8 \cdot \frac{1}{t}\right)}\right)} \]

      4. associate--r+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{{t}^{3}}\right) - 8 \cdot \frac{1}{t}\right)}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      7. associate-/r* N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - 16 \cdot \color{blue}{\frac{\frac{1}{t}}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\left(\frac{12}{{t}^{2}} - \color{blue}{\frac{16 \cdot \frac{1}{t}}{{t}^{2}}}\right) - 8 \cdot \frac{1}{t}\right)} \]

      9. div-sub N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{12 - 16 \cdot \frac{1}{t}}{{t}^{2}}} - 8 \cdot \frac{1}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12 - 16 \cdot \frac{1}{t}}{\color{blue}{t \cdot t}} - 8 \cdot \frac{1}{t}\right)} \]

      11. associate-/l/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}} - 8 \cdot \frac{1}{t}\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t}}{t} - \frac{\color{blue}{8}}{t}\right)} \]

      14. div-sub N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\frac{\frac{12 - 16 \cdot \frac{1}{t}}{t} - 8}{t}}} \]

   5. Simplified 99.8%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \frac{-8 + \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   6. Taylor expanded in t around -inf

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

      1. +-lowering-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. /-lowering-/.f64 N/A

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

   8. Simplified 99.9%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   9. 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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.5

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

   10. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (+
    0.8333333333333334
    (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t)))
   (fma (* t t) (fma t (+ t -2.0) 1.0) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	} else {
		tmp = fma((t * t), fma(t, (t + -2.0), 1.0), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.001)
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	else
		tmp = fma(Float64(t * t), fma(t, Float64(t + -2.0), 1.0), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] * N[(t * N[(t + -2.0), $MachinePrecision] + 1.0), $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))) < 1e-3

   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

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) - 8 \cdot \frac{1}{t}}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12}{\color{blue}{t \cdot t}} - 8 \cdot \frac{1}{t}\right)} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\color{blue}{\frac{\frac{12}{t}}{t}} - 8 \cdot \frac{1}{t}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t} - 8 \cdot \frac{1}{t}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{\color{blue}{12 \cdot \frac{1}{t}}}{t} - 8 \cdot \frac{1}{t}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12 \cdot \frac{1}{t}}{t} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \left(\frac{12 \cdot \frac{1}{t}}{t} - \frac{\color{blue}{8}}{t}\right)} \]

      9. div-sub N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\frac{12 \cdot \frac{1}{t} - 8}{t}}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \color{blue}{\frac{12 \cdot \frac{1}{t} - 8}{t}}} \]

      11. sub-neg N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{\color{blue}{12 \cdot \frac{1}{t} + \left(\mathsf{neg}\left(8\right)\right)}}{t}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{12 \cdot \frac{1}{t} + \color{blue}{-8}}{t}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{\color{blue}{-8 + 12 \cdot \frac{1}{t}}}{t}} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{\color{blue}{-8 + 12 \cdot \frac{1}{t}}}{t}} \]

      15. associate-*r/ N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{-8 + \color{blue}{\frac{12 \cdot 1}{t}}}{t}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{-8 + \frac{\color{blue}{12}}{t}}{t}} \]

      17. /-lowering-/.f64 99.7

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 + \frac{-8 + \color{blue}{\frac{12}{t}}}{t}} \]

   5. Simplified 99.7%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 + \frac{-8 + \frac{12}{t}}{t}}} \]

   6. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{5}{6} + \left(\color{blue}{\left(0 - \frac{2}{9} \cdot \frac{1}{t}\right)} + \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      6. associate--r- N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\left(0 - \left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{5}{6} + \left(0 - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \frac{\frac{1}{27}}{{t}^{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{5}{6} + \left(0 - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{5}{6} + \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{\color{blue}{t \cdot t}}\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \frac{5}{6} + \left(0 - \left(\frac{\frac{2}{9}}{t} - \color{blue}{\frac{\frac{\frac{1}{27}}{t}}{t}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{5}{6} + \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\color{blue}{\frac{1}{27} \cdot 1}}{t}}{t}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \frac{5}{6} + \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\color{blue}{\frac{1}{27} \cdot \frac{1}{t}}}{t}\right)\right) \]

      13. div-sub N/A

          \[\leadsto \frac{5}{6} + \left(0 - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}}\right) \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 99.8

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

   11. Simplified 99.8%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   9. 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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.5

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

   10. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (* t t) (fma t (+ t -2.0) 1.0) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma((t * t), fma(t, (t + -2.0), 1.0), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.001)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(Float64(t * t), fma(t, Float64(t + -2.0), 1.0), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] * N[(t * N[(t + -2.0), $MachinePrecision] + 1.0), $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))) < 1e-3

   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

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf

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

      1. --lowering--.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. /-lowering-/.f64 99.6

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

   8. Simplified 99.6%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   9. 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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.5

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

   10. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t + -2, 1\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma t (fma -2.0 (* t t) t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(t, fma(-2.0, (t * t), t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 0.001)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(t, fma(-2.0, Float64(t * t), t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(t * N[(-2.0 * N[(t * t), $MachinePrecision] + 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))) < 1e-3

   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

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf

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

      1. --lowering--.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. /-lowering-/.f64 99.6

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

   8. Simplified 99.6%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 99.5

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

   10. Simplified 99.5%

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

Alternative 9: 99.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		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(1.0 + Float64(1.0 / t))) <= 0.001)
		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[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], 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))) < 1e-3

   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

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf

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

      1. --lowering--.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. /-lowering-/.f64 99.6

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

   8. Simplified 99.6%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   9. 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. accelerator-lowering-fma.f64 99.5

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

   10. Simplified 99.5%

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

Alternative 10: 98.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 0.001:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 0.001)
   0.8333333333333334
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 0.001) {
		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(1.0 + Float64(1.0 / t))) <= 0.001)
		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[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.001], 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))) < 1e-3

   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

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 98.7%

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

   if 1e-3 < (/.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   9. 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. accelerator-lowering-fma.f64 99.5

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

   10. Simplified 99.5%

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

Alternative 11: 98.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 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 + (1.0d0 / t))) <= 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 + (1.0 / t))) <= 1.0) {
		tmp = 0.8333333333333334;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 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(1.0 + Float64(1.0 / t))) <= 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 + (1.0 / t))) <= 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[(1.0 + N[(1.0 / t), $MachinePrecision]), $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 inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 98.7%

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

   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

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

   5. Simplified 99.3%

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

   6. Taylor expanded in t around 0

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

   8. Simplified 99.3%

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

Alternative 12: 60.0% 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

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

5. Simplified 62.3%

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

6. Taylor expanded in t around 0

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

8. Simplified 63.2%

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

Reproduce

herbie shell --seed 2024199 
(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))))))))