Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 13

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

   4. fabs-sqr-rev N/A

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

   5. sqr-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

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

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

6. Applied rewrites 100.0%

   \[\leadsto 1 - \sqrt{{\color{blue}{\left(\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)\right)}}^{-2}} \]
7. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
     t))
   (-
    1.0
    (pow
     (+ 2.0 (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t))
     -1.0))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / t);
	} else {
		tmp = 1.0 - pow((2.0 + ((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t)), -1.0);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 0.005)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t));
	else
		tmp = Float64(1.0 - (Float64(2.0 + Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t)) ^ -1.0));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(2.0 + N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], -1.0], $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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l* N/A

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

      11. div-add N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. div-sub N/A

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

   5. Applied rewrites 99.4%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), t \cdot t, 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
     t))
   (- 1.0 (sqrt (fma (fma (fma -8.0 (* t t) 2.0) t -1.0) (* t t) 0.25)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / t);
	} else {
		tmp = 1.0 - sqrt(fma(fma(fma(-8.0, (t * t), 2.0), t, -1.0), (t * t), 0.25));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 0.005)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t));
	else
		tmp = Float64(1.0 - sqrt(fma(fma(fma(-8.0, Float64(t * t), 2.0), t, -1.0), Float64(t * t), 0.25)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(N[(-8.0 * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * t + -1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.25), $MachinePrecision]], $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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l* N/A

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

      11. div-add N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. div-sub N/A

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

   5. Applied rewrites 99.4%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr-rev N/A

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

      5. sqr-pow N/A

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

      6. inv-pow N/A

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

      7. lift-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.6

          \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), \color{blue}{t \cdot t}, 0.25\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), t \cdot t, 0.25\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), t \cdot t, 0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), t \cdot t, 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   (-
    0.8333333333333334
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))
   (- 1.0 (sqrt (fma (fma (fma -8.0 (* t t) 2.0) t -1.0) (* t t) 0.25)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	} else {
		tmp = 1.0 - sqrt(fma(fma(fma(-8.0, (t * t), 2.0), t, -1.0), (t * t), 0.25));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 0.005)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	else
		tmp = Float64(1.0 - sqrt(fma(fma(fma(-8.0, Float64(t * t), 2.0), t, -1.0), Float64(t * t), 0.25)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(N[(-8.0 * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * t + -1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.25), $MachinePrecision]], $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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. div-sub N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr-rev N/A

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

      5. sqr-pow N/A

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

      6. inv-pow N/A

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

      7. lift-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.6

          \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), \color{blue}{t \cdot t}, 0.25\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), t \cdot t, 0.25\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-8, t \cdot t, 2\right), t, -1\right), t \cdot t, 0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0)))) 0.005)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (-
    0.8333333333333334
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)))) <= 0.005) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0)))) <= 0.005)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.0050000000000000001 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. sub-neg N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. div-sub N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0)))) 0.005)
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)))) <= 0.005) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0)))) <= 0.005)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.0050000000000000001 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.1

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222}{t}}\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right) \cdot t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))
   (fma (* (fma -2.0 t 1.0) t) t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	} else {
		tmp = fma((fma(-2.0, t, 1.0) * t), t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 0.005)
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	else
		tmp = fma(Float64(fma(-2.0, t, 1.0) * t), t, 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * t + 1.0), $MachinePrecision] * t), $MachinePrecision] * 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.1

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222}{t}}\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right) \cdot t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right) \cdot t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (* (fma -2.0 t 1.0) t) t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma((fma(-2.0, t, 1.0) * t), t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))) <= 0.005)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(Float64(fma(-2.0, t, 1.0) * t), t, 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * t + 1.0), $MachinePrecision] * t), $MachinePrecision] * 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right) \cdot t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		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 + (t ^ -1.0))) <= 0.005)
		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[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. sqr-abs-rev N/A

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

   3. lift--.f64 N/A

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

   4. fabs-sub N/A

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

   5. lift--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sqr-abs N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 11: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))) 0.005)
   0.8333333333333334
   (fma t t 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + pow(t, -1.0))) <= 0.005) {
		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 + (t ^ -1.0))) <= 0.005)
		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[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], 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))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.4%

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

   if 0.0050000000000000001 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + {t}^{-1}} \leq 0.005:\\ \;\;\;\;0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t):
	tmp = 0
	if (2.0 - ((2.0 / t) / (1.0 + math.pow(t, -1.0)))) <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

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

Alternative 13: 58.8% accurate, 101.0× 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%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 58.0%

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

Reproduce

herbie shell --seed 2024305 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))