Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 21.3s

Alternatives: 11

Speedup: 1.6×

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ t_2 := t\_1 \cdot \left(t\_1 + -4\right)\\ \frac{5 + t\_2}{t\_2 + 6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))) (t_2 (* t_1 (+ t_1 -4.0))))
   (/ (+ 5.0 t_2) (+ t_2 6.0))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = t_1 * (t_1 + -4.0);
	return (5.0 + t_2) / (t_2 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 / (1.0d0 + t)
    t_2 = t_1 * (t_1 + (-4.0d0))
    code = (5.0d0 + t_2) / (t_2 + 6.0d0)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	double t_2 = t_1 * (t_1 + -4.0);
	return (5.0 + t_2) / (t_2 + 6.0);
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	t_2 = t_1 * (t_1 + -4.0)
	return (5.0 + t_2) / (t_2 + 6.0)

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	t_2 = Float64(t_1 * Float64(t_1 + -4.0))
	return Float64(Float64(5.0 + t_2) / Float64(t_2 + 6.0))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	t_2 = t_1 * (t_1 + -4.0);
	tmp = (5.0 + t_2) / (t_2 + 6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$2), $MachinePrecision] / N[(t$95$2 + 6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;\frac{-0.2222222222222222}{t} + \left(0.8333333333333334 - \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{0.04938271604938271}{t \cdot \left(t \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.52)
   (+
    (/ -0.2222222222222222 t)
    (-
     0.8333333333333334
     (/ (+ -0.037037037037037035 (/ -0.04938271604938271 t)) (* t t))))
   (if (<= t 0.58)
     (+ 0.5 (* t (* t (+ 1.0 (* t -2.0)))))
     (+
      (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t)
      (+ 0.8333333333333334 (/ 0.04938271604938271 (* t (* t t))))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)));
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = ((-0.2222222222222222 + (0.037037037037037035 / t)) / t) + (0.8333333333333334 + (0.04938271604938271 / (t * (t * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.52d0)) then
        tmp = ((-0.2222222222222222d0) / t) + (0.8333333333333334d0 - (((-0.037037037037037035d0) + ((-0.04938271604938271d0) / t)) / (t * t)))
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t) + (0.8333333333333334d0 + (0.04938271604938271d0 / (t * (t * t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)));
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = ((-0.2222222222222222 + (0.037037037037037035 / t)) / t) + (0.8333333333333334 + (0.04938271604938271 / (t * (t * t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.52:
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)))
	elif t <= 0.58:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = ((-0.2222222222222222 + (0.037037037037037035 / t)) / t) + (0.8333333333333334 + (0.04938271604938271 / (t * (t * t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(Float64(-0.2222222222222222 / t) + Float64(0.8333333333333334 - Float64(Float64(-0.037037037037037035 + Float64(-0.04938271604938271 / t)) / Float64(t * t))));
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = Float64(Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t) + Float64(0.8333333333333334 + Float64(0.04938271604938271 / Float64(t * Float64(t * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.52)
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)));
	elseif (t <= 0.58)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = ((-0.2222222222222222 + (0.037037037037037035 / t)) / t) + (0.8333333333333334 + (0.04938271604938271 / (t * (t * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.52], N[(N[(-0.2222222222222222 / t), $MachinePrecision] + N[(0.8333333333333334 - N[(N[(-0.037037037037037035 + N[(-0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.58], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(0.8333333333333334 + N[(0.04938271604938271 / N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{27}}{{t}^{2}} - \frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)}\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in t around -inf

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right), \color{blue}{t}\right)\right) \]

   10. Simplified 99.7%

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. associate-+l- N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-2}{9}}{t}\right), \color{blue}{\left(\frac{\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t}}{t} - \frac{5}{6}\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \left(\color{blue}{\frac{\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t}}{t}} - \frac{5}{6}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\left(\frac{\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t}}{t}\right), \color{blue}{\frac{5}{6}}\right)\right) \]

       7. associate-/l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t \cdot t}\right), \frac{5}{6}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{27} + \frac{\frac{-4}{81}}{t}\right), \left(t \cdot t\right)\right), \frac{5}{6}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \left(\frac{\frac{-4}{81}}{t}\right)\right), \left(t \cdot t\right)\right), \frac{5}{6}\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{/.f64}\left(\frac{-4}{81}, t\right)\right), \left(t \cdot t\right)\right), \frac{5}{6}\right)\right) \]

       11. *-lowering-*.f64 99.7%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{/.f64}\left(\frac{-4}{81}, t\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \frac{5}{6}\right)\right) \]

   12. Applied egg-rr 99.7%

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

   if -0.52000000000000002 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   if 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{27}}{{t}^{2}} - \frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t} + \left(0.8333333333333334 + \frac{0.04938271604938271}{t \cdot \left(t \cdot t\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;\frac{-0.2222222222222222}{t} + \left(0.8333333333333334 - \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t} + \left(0.8333333333333334 + \frac{0.04938271604938271}{t \cdot \left(t \cdot t\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;\frac{-0.2222222222222222}{t} + \left(0.8333333333333334 - \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.52)
   (+
    (/ -0.2222222222222222 t)
    (-
     0.8333333333333334
     (/ (+ -0.037037037037037035 (/ -0.04938271604938271 t)) (* t t))))
   (if (<= t 0.58)
     (+ 0.5 (* t (* t (+ 1.0 (* t -2.0)))))
     (+
      0.8333333333333334
      (/
       (+
        -0.2222222222222222
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)));
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.52d0)) then
        tmp = ((-0.2222222222222222d0) / t) + (0.8333333333333334d0 - (((-0.037037037037037035d0) + ((-0.04938271604938271d0) / t)) / (t * t)))
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = 0.8333333333333334d0 + (((-0.2222222222222222d0) + ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)));
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.52:
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)))
	elif t <= 0.58:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(Float64(-0.2222222222222222 / t) + Float64(0.8333333333333334 - Float64(Float64(-0.037037037037037035 + Float64(-0.04938271604938271 / t)) / Float64(t * t))));
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.52)
		tmp = (-0.2222222222222222 / t) + (0.8333333333333334 - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t)));
	elseif (t <= 0.58)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.52], N[(N[(-0.2222222222222222 / t), $MachinePrecision] + N[(0.8333333333333334 - N[(N[(-0.037037037037037035 + N[(-0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.58], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{27}}{{t}^{2}} - \frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)}\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in t around -inf

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right), \color{blue}{t}\right)\right) \]

   10. Simplified 99.7%

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

       1. +-commutative N/A

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

       2. div-sub N/A

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

       3. associate-+l- N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-2}{9}}{t}\right), \color{blue}{\left(\frac{\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t}}{t} - \frac{5}{6}\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \left(\color{blue}{\frac{\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t}}{t}} - \frac{5}{6}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\left(\frac{\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t}}{t}\right), \color{blue}{\frac{5}{6}}\right)\right) \]

       7. associate-/l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{27} + \frac{\frac{-4}{81}}{t}}{t \cdot t}\right), \frac{5}{6}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{27} + \frac{\frac{-4}{81}}{t}\right), \left(t \cdot t\right)\right), \frac{5}{6}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \left(\frac{\frac{-4}{81}}{t}\right)\right), \left(t \cdot t\right)\right), \frac{5}{6}\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{/.f64}\left(\frac{-4}{81}, t\right)\right), \left(t \cdot t\right)\right), \frac{5}{6}\right)\right) \]

       11. *-lowering-*.f64 99.7%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-2}{9}, t\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{/.f64}\left(\frac{-4}{81}, t\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \frac{5}{6}\right)\right) \]

   12. Applied egg-rr 99.7%

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

   if -0.52000000000000002 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   if 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \frac{2}{9}\right)\right)}{t}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}{t}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}\right), \color{blue}{t}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;\frac{-0.2222222222222222}{t} + \left(0.8333333333333334 - \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (+
            -0.2222222222222222
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t))
           t))))
   (if (<= t -0.52)
     t_1
     (if (<= t 0.58) (+ 0.5 (* t (* t (+ 1.0 (* t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	double tmp;
	if (t <= -0.52) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + (((-0.2222222222222222d0) + ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t)) / t)
    if (t <= (-0.52d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	double tmp;
	if (t <= -0.52) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t)
	tmp = 0
	if t <= -0.52:
		tmp = t_1
	elif t <= 0.58:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t)) / t))
	tmp = 0.0
	if (t <= -0.52)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + ((0.037037037037037035 + (0.04938271604938271 / t)) / t)) / t);
	tmp = 0.0;
	if (t <= -0.52)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.52], t$95$1, If[LessEqual[t, 0.58], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \frac{2}{9}\right)\right)}{t}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}{t}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}\right), \color{blue}{t}\right)\right) \]

   7. Simplified 99.8%

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

   if -0.52000000000000002 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{-0.2222222222222222}{t} + 0.8333333333333334\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          (+ (/ -0.2222222222222222 t) 0.8333333333333334)
          (/ 0.037037037037037035 (* t t)))))
   (if (<= t -0.62)
     t_1
     (if (<= t 0.44) (+ 0.5 (* t (* t (+ 1.0 (* t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = ((-0.2222222222222222 / t) + 0.8333333333333334) + (0.037037037037037035 / (t * t));
	double tmp;
	if (t <= -0.62) {
		tmp = t_1;
	} else if (t <= 0.44) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((-0.2222222222222222d0) / t) + 0.8333333333333334d0) + (0.037037037037037035d0 / (t * t))
    if (t <= (-0.62d0)) then
        tmp = t_1
    else if (t <= 0.44d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = ((-0.2222222222222222 / t) + 0.8333333333333334) + (0.037037037037037035 / (t * t));
	double tmp;
	if (t <= -0.62) {
		tmp = t_1;
	} else if (t <= 0.44) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((-0.2222222222222222 / t) + 0.8333333333333334) + (0.037037037037037035 / (t * t))
	tmp = 0
	if t <= -0.62:
		tmp = t_1
	elif t <= 0.44:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(-0.2222222222222222 / t) + 0.8333333333333334) + Float64(0.037037037037037035 / Float64(t * t)))
	tmp = 0.0
	if (t <= -0.62)
		tmp = t_1;
	elseif (t <= 0.44)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = ((-0.2222222222222222 / t) + 0.8333333333333334) + (0.037037037037037035 / (t * t));
	tmp = 0.0;
	if (t <= -0.62)
		tmp = t_1;
	elseif (t <= 0.44)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(-0.2222222222222222 / t), $MachinePrecision] + 0.8333333333333334), $MachinePrecision] + N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.62], t$95$1, If[LessEqual[t, 0.44], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.619999999999999996 or 0.440000000000000002 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{27}}{{t}^{2}} - \frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)}\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)}\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{5}{6} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right), \left(\color{blue}{\frac{\frac{1}{27}}{{t}^{2}}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right), \left(\color{blue}{\frac{\frac{1}{27}}{{t}^{2}}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right), \left(\frac{\frac{1}{27}}{{\color{blue}{t}}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right), \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{t}\right)\right), \left(\frac{\frac{1}{27}}{\color{blue}{{t}^{2}}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{-2}{9}}{t}\right)\right), \left(\frac{\frac{1}{27}}{{\color{blue}{t}}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{\frac{1}{27}}{\color{blue}{{t}^{2}}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{\frac{1}{27} \cdot 1}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{1}{27} \cdot \frac{1}{{t}^{2}} + \color{blue}{\frac{4}{81}} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      14. unpow3 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{1}{27} \cdot \frac{1}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{1}{27} \cdot \frac{1}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{2} \cdot t}\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{1}{27} \cdot \frac{1}{{t}^{2}} + \frac{4}{81} \cdot \frac{\frac{1}{{t}^{2}}}{\color{blue}{t}}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{1}{27} \cdot \frac{1}{{t}^{2}} + \frac{\frac{4}{81} \cdot \frac{1}{{t}^{2}}}{\color{blue}{t}}\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \left(\frac{1}{27} \cdot \frac{1}{{t}^{2}} + \frac{\frac{4}{81}}{t} \cdot \color{blue}{\frac{1}{{t}^{2}}}\right)\right) \]

   10. Simplified 99.8%

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

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \mathsf{/.f64}\left(\frac{1}{27}, \color{blue}{\left({t}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \mathsf{/.f64}\left(\frac{1}{27}, \left(t \cdot \color{blue}{t}\right)\right)\right) \]

       3. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right), \mathsf{/.f64}\left(\frac{1}{27}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   13. Simplified 99.6%

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

   if -0.619999999999999996 < t < 0.440000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;\left(\frac{-0.2222222222222222}{t} + 0.8333333333333334\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.2222222222222222}{t} + 0.8333333333333334\right) + \frac{0.037037037037037035}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.62)
     t_1
     (if (<= t 0.44) (+ 0.5 (* t (* t (+ 1.0 (* t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.62) {
		tmp = t_1;
	} else if (t <= 0.44) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    if (t <= (-0.62d0)) then
        tmp = t_1
    else if (t <= 0.44d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.62) {
		tmp = t_1;
	} else if (t <= 0.44) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.62:
		tmp = t_1
	elif t <= 0.44:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.62)
		tmp = t_1;
	elseif (t <= 0.44)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.62)
		tmp = t_1;
	elseif (t <= 0.44)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.62], t$95$1, If[LessEqual[t, 0.44], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.619999999999999996 or 0.440000000000000002 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} - \frac{2}{9} \cdot \frac{1}{t}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{1}{27}}{{t}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) + \color{blue}{\frac{\frac{1}{27}}{{t}^{2}}}\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\left(0 - \frac{2}{9} \cdot \frac{1}{t}\right) + \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right)\right) \]

      6. associate--r- N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \color{blue}{\left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9} \cdot 1}{t} - \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{t \cdot \color{blue}{t}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27}}{t}}{\color{blue}{t}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27} \cdot 1}{t}}{t}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)\right) \]

      13. div-sub N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{t}}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)\right) \]

      15. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}\right)\right)}{\color{blue}{t}}\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + \left(\mathsf{neg}\left(\frac{1}{27} \cdot \frac{1}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{27} \cdot \frac{1}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{1}{27} \cdot \frac{1}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)\right) \]

   7. Simplified 99.6%

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

   if -0.619999999999999996 < t < 0.440000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.82) t_1 (if (<= t 0.235) (+ (* t t) 0.5) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    if (t <= (-0.82d0)) then
        tmp = t_1
    else if (t <= 0.235d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.82:
		tmp = t_1
	elif t <= 0.235:
		tmp = (t * t) + 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.235)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.235)
		tmp = (t * t) + 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.82], t$95$1, If[LessEqual[t, 0.235], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.23499999999999999 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} - \frac{2}{9} \cdot \frac{1}{t}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{1}{27}}{{t}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) + \color{blue}{\frac{\frac{1}{27}}{{t}^{2}}}\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\left(0 - \frac{2}{9} \cdot \frac{1}{t}\right) + \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right)\right) \]

      6. associate--r- N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \color{blue}{\left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9} \cdot 1}{t} - \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{t \cdot \color{blue}{t}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27}}{t}}{\color{blue}{t}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27} \cdot 1}{t}}{t}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)\right) \]

      13. div-sub N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(0 - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{t}}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)\right) \]

      15. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}\right)\right)}{\color{blue}{t}}\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + \left(\mathsf{neg}\left(\frac{1}{27} \cdot \frac{1}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{27} \cdot \frac{1}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{1}{27} \cdot \frac{1}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)\right) \]

   7. Simplified 99.6%

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

   if -0.819999999999999951 < t < 0.23499999999999999

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.2222222222222222}{t} + 0.8333333333333334\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ (/ -0.2222222222222222 t) 0.8333333333333334)))
   (if (<= t -0.8) t_1 (if (<= t 0.58) (+ (* t t) 0.5) t_1))))

C

double code(double t) {
	double t_1 = (-0.2222222222222222 / t) + 0.8333333333333334;
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-0.2222222222222222d0) / t) + 0.8333333333333334d0
    if (t <= (-0.8d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (-0.2222222222222222 / t) + 0.8333333333333334;
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (-0.2222222222222222 / t) + 0.8333333333333334
	tmp = 0
	if t <= -0.8:
		tmp = t_1
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(-0.2222222222222222 / t) + 0.8333333333333334)
	tmp = 0.0
	if (t <= -0.8)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (-0.2222222222222222 / t) + 0.8333333333333334;
	tmp = 0.0;
	if (t <= -0.8)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-0.2222222222222222 / t), $MachinePrecision] + 0.8333333333333334), $MachinePrecision]}, If[LessEqual[t, -0.8], t$95$1, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 99.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 99.2%

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

   if -0.80000000000000004 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;\frac{-0.2222222222222222}{t} + 0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.2222222222222222}{t} + 0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.58) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.9], 0.8333333333333334, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

   7. Simplified 98.4%

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

   if -0.900000000000000022 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

   7. Simplified 98.4%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Simplified 99.7%

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

Alternative 11: 59.6% accurate, 51.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0

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

7. Simplified 58.4%

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

Reproduce

herbie shell --seed 2024141 
(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))))))))