Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 12

Speedup: 1.9×

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

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ t 1.0))) (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 / (t + 1.0);
	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 / (t + 1.0d0)
    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 / (t + 1.0);
	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 / (t + 1.0)
	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(t + 1.0))
	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 / (t + 1.0);
	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[(t + 1.0), $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%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-8 + \frac{4}{t + 1}}{t + 1}\\ \frac{5 + t\_1}{6 + t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ -8.0 (/ 4.0 (+ t 1.0))) (+ t 1.0))))
   (/ (+ 5.0 t_1) (+ 6.0 t_1))))

C

double code(double t) {
	double t_1 = (-8.0 + (4.0 / (t + 1.0))) / (t + 1.0);
	return (5.0 + t_1) / (6.0 + t_1);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((-8.0d0) + (4.0d0 / (t + 1.0d0))) / (t + 1.0d0)
    code = (5.0d0 + t_1) / (6.0d0 + t_1)
end function

Java

public static double code(double t) {
	double t_1 = (-8.0 + (4.0 / (t + 1.0))) / (t + 1.0);
	return (5.0 + t_1) / (6.0 + t_1);
}

Python

def code(t):
	t_1 = (-8.0 + (4.0 / (t + 1.0))) / (t + 1.0)
	return (5.0 + t_1) / (6.0 + t_1)

Julia

function code(t)
	t_1 = Float64(Float64(-8.0 + Float64(4.0 / Float64(t + 1.0))) / Float64(t + 1.0))
	return Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1))
end

MATLAB

function tmp = code(t)
	t_1 = (-8.0 + (4.0 / (t + 1.0))) / (t + 1.0);
	tmp = (5.0 + t_1) / (6.0 + t_1);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-8.0 + N[(4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $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. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. flip3-+ N/A

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

   3. associate-/r/ N/A

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

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

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

6. Applied egg-rr 78.1%

   \[\leadsto \frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \color{blue}{\frac{-8 + 2 \cdot \frac{2}{1 + t}}{1 + t \cdot \left(t \cdot t\right)} \cdot \left(1 + t \cdot \left(t - 1\right)\right)}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{-8 + \frac{4}{1 + t}}{1 + t}}{6 + \frac{-8 + \frac{4}{1 + t}}{1 + t}}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{5 + \frac{-8 + \frac{4}{t + 1}}{t + 1}}{6 + \frac{-8 + \frac{4}{t + 1}}{t + 1}} \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1.2 + \frac{\frac{0.032 - \frac{0.0768}{t}}{t} - -0.32}{t}}\\ \mathbf{if}\;t \leq -0.45:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.62:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.2 (/ (- (/ (- 0.032 (/ 0.0768 t)) t) -0.32) t)))))
   (if (<= t -0.45)
     t_1
     (if (<= t 0.62) (+ 0.5 (* t (* t (+ (* t (+ t -2.0)) 1.0)))) t_1))))

C

double code(double t) {
	double t_1 = 1.0 / (1.2 + ((((0.032 - (0.0768 / t)) / t) - -0.32) / t));
	double tmp;
	if (t <= -0.45) {
		tmp = t_1;
	} else if (t <= 0.62) {
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	} 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 = 1.0d0 / (1.2d0 + ((((0.032d0 - (0.0768d0 / t)) / t) - (-0.32d0)) / t))
    if (t <= (-0.45d0)) then
        tmp = t_1
    else if (t <= 0.62d0) then
        tmp = 0.5d0 + (t * (t * ((t * (t + (-2.0d0))) + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 1.0 / (1.2 + ((((0.032 - (0.0768 / t)) / t) - -0.32) / t));
	double tmp;
	if (t <= -0.45) {
		tmp = t_1;
	} else if (t <= 0.62) {
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 1.0 / (1.2 + ((((0.032 - (0.0768 / t)) / t) - -0.32) / t))
	tmp = 0
	if t <= -0.45:
		tmp = t_1
	elif t <= 0.62:
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(1.0 / Float64(1.2 + Float64(Float64(Float64(Float64(0.032 - Float64(0.0768 / t)) / t) - -0.32) / t)))
	tmp = 0.0
	if (t <= -0.45)
		tmp = t_1;
	elseif (t <= 0.62)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(Float64(t * Float64(t + -2.0)) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 1.0 / (1.2 + ((((0.032 - (0.0768 / t)) / t) - -0.32) / t));
	tmp = 0.0;
	if (t <= -0.45)
		tmp = t_1;
	elseif (t <= 0.62)
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 / N[(1.2 + N[(N[(N[(N[(0.032 - N[(0.0768 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - -0.32), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.45], t$95$1, If[LessEqual[t, 0.62], N[(0.5 + N[(t * N[(t * N[(N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.450000000000000011 or 0.619999999999999996 < 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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{6}{5} + \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{6}{5} - \color{blue}{\frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \color{blue}{\left(\frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\left(-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}\right), \color{blue}{t}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\left(-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{8}{25}\right)\right)\right), t\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{8}{25}\right)\right) + -1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}\right), t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{8}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}\right)\right)\right), t\right)\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{8}{25}\right)\right) - \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}\right), t\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{8}{25}\right)\right), \left(\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}\right)\right), t\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-8}{25}, \left(\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}\right)\right), t\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-8}{25}, \mathsf{/.f64}\left(\left(\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}\right), t\right)\right), t\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-8}{25}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{4}{125}, \left(\frac{48}{625} \cdot \frac{1}{t}\right)\right), t\right)\right), t\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-8}{25}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{4}{125}, \left(\frac{\frac{48}{625} \cdot 1}{t}\right)\right), t\right)\right), t\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-8}{25}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{4}{125}, \left(\frac{\frac{48}{625}}{t}\right)\right), t\right)\right), t\right)\right)\right) \]

      15. /-lowering-/.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-8}{25}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{4}{125}, \mathsf{/.f64}\left(\frac{48}{625}, t\right)\right), t\right)\right), t\right)\right)\right) \]

   9. Simplified 98.6%

      \[\leadsto \frac{1}{\color{blue}{1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}}} \]

   if -0.450000000000000011 < t < 0.619999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.45:\\ \;\;\;\;\frac{1}{1.2 + \frac{\frac{0.032 - \frac{0.0768}{t}}{t} - -0.32}{t}}\\ \mathbf{elif}\;t \leq 0.62:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + \frac{\frac{0.032 - \frac{0.0768}{t}}{t} - -0.32}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{\frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t} + 0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (-
          0.8333333333333334
          (/
           (+
            (/ (+ -0.037037037037037035 (/ -0.04938271604938271 t)) t)
            0.2222222222222222)
           t))))
   (if (<= t -0.48)
     t_1
     (if (<= t 0.75) (+ 0.5 (* t (* t (+ (* t (+ t -2.0)) 1.0)))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - ((((-0.037037037037037035 + (-0.04938271604938271 / t)) / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.48) {
		tmp = t_1;
	} else if (t <= 0.75) {
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	} 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 - (((((-0.037037037037037035d0) + ((-0.04938271604938271d0) / t)) / t) + 0.2222222222222222d0) / t)
    if (t <= (-0.48d0)) then
        tmp = t_1
    else if (t <= 0.75d0) then
        tmp = 0.5d0 + (t * (t * ((t * (t + (-2.0d0))) + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - ((((-0.037037037037037035 + (-0.04938271604938271 / t)) / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.48) {
		tmp = t_1;
	} else if (t <= 0.75) {
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - ((((-0.037037037037037035 + (-0.04938271604938271 / t)) / t) + 0.2222222222222222) / t)
	tmp = 0
	if t <= -0.48:
		tmp = t_1
	elif t <= 0.75:
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(Float64(Float64(Float64(-0.037037037037037035 + Float64(-0.04938271604938271 / t)) / t) + 0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.48)
		tmp = t_1;
	elseif (t <= 0.75)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(Float64(t * Float64(t + -2.0)) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - ((((-0.037037037037037035 + (-0.04938271604938271 / t)) / t) + 0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.48)
		tmp = t_1;
	elseif (t <= 0.75)
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(N[(N[(N[(-0.037037037037037035 + N[(-0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.48], t$95$1, If[LessEqual[t, 0.75], N[(0.5 + N[(t * N[(t * N[(N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.75 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   5. Simplified 98.6%

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

   if -0.47999999999999998 < t < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t} + 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t} + 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} + 0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (-
          0.8333333333333334
          (/ (+ (/ -0.037037037037037035 t) 0.2222222222222222) t))))
   (if (<= t -0.56)
     t_1
     (if (<= t 0.6) (+ 0.5 (* t (* t (+ (* t (+ t -2.0)) 1.0)))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.56) {
		tmp = t_1;
	} else if (t <= 0.6) {
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	} 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 - ((((-0.037037037037037035d0) / t) + 0.2222222222222222d0) / t)
    if (t <= (-0.56d0)) then
        tmp = t_1
    else if (t <= 0.6d0) then
        tmp = 0.5d0 + (t * (t * ((t * (t + (-2.0d0))) + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.56) {
		tmp = t_1;
	} else if (t <= 0.6) {
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t)
	tmp = 0
	if t <= -0.56:
		tmp = t_1
	elif t <= 0.6:
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(Float64(Float64(-0.037037037037037035 / t) + 0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.56)
		tmp = t_1;
	elseif (t <= 0.6)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(Float64(t * Float64(t + -2.0)) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.56)
		tmp = t_1;
	elseif (t <= 0.6)
		tmp = 0.5 + (t * (t * ((t * (t + -2.0)) + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(N[(N[(-0.037037037037037035 / t), $MachinePrecision] + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.56], t$95$1, If[LessEqual[t, 0.6], N[(0.5 + N[(t * N[(t * N[(N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.56000000000000005 or 0.599999999999999978 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

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

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

      14. sub-neg N/A

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

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

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.56000000000000005 < t < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} + 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} + 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1;
	} else if (t <= 0.45) {
		tmp = 0.5 + (t * (t * ((t * -2.0) + 1.0)));
	} 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 - ((((-0.037037037037037035d0) / t) + 0.2222222222222222d0) / t)
    if (t <= (-0.6d0)) then
        tmp = t_1
    else if (t <= 0.45d0) then
        tmp = 0.5d0 + (t * (t * ((t * (-2.0d0)) + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.450000000000000011 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

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

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

      14. sub-neg N/A

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

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

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.599999999999999978 < t < 0.450000000000000011

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. 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 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.0%

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

Alternative 7: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} + 0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (-
          0.8333333333333334
          (/ (+ (/ -0.037037037037037035 t) 0.2222222222222222) t))))
   (if (<= t -0.82) t_1 (if (<= t 0.33) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.33) {
		tmp = 0.5 + (t * t);
	} 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 - ((((-0.037037037037037035d0) / t) + 0.2222222222222222d0) / t)
    if (t <= (-0.82d0)) then
        tmp = t_1
    else if (t <= 0.33d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.33) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t)
	tmp = 0
	if t <= -0.82:
		tmp = t_1
	elif t <= 0.33:
		tmp = 0.5 + (t * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 - Float64(Float64(Float64(-0.037037037037037035 / t) + 0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.33)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 - (((-0.037037037037037035 / t) + 0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.33)
		tmp = 0.5 + (t * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 - N[(N[(N[(-0.037037037037037035 / t), $MachinePrecision] + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.82], t$95$1, If[LessEqual[t, 0.33], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.330000000000000016 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

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

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

      14. sub-neg N/A

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

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

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.819999999999999951 < t < 0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. 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 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} + 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} + 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{if}\;t \leq -0.85:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.39:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.2 (/ 0.32 t)))))
   (if (<= t -0.85) t_1 (if (<= t 0.39) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 1.0 / (1.2 + (0.32 / t));
	double tmp;
	if (t <= -0.85) {
		tmp = t_1;
	} else if (t <= 0.39) {
		tmp = 0.5 + (t * t);
	} 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 = 1.0d0 / (1.2d0 + (0.32d0 / t))
    if (t <= (-0.85d0)) then
        tmp = t_1
    else if (t <= 0.39d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 1.0 / (1.2 + (0.32 / t));
	double tmp;
	if (t <= -0.85) {
		tmp = t_1;
	} else if (t <= 0.39) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 1.0 / (1.2 + (0.32 / t))
	tmp = 0
	if t <= -0.85:
		tmp = t_1
	elif t <= 0.39:
		tmp = 0.5 + (t * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(1.0 / Float64(1.2 + Float64(0.32 / t)))
	tmp = 0.0
	if (t <= -0.85)
		tmp = t_1;
	elseif (t <= 0.39)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 1.0 / (1.2 + (0.32 / t));
	tmp = 0.0;
	if (t <= -0.85)
		tmp = t_1;
	elseif (t <= 0.39)
		tmp = 0.5 + (t * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 / N[(1.2 + N[(0.32 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.85], t$95$1, If[LessEqual[t, 0.39], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.849999999999999978 or 0.39000000000000001 < 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. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{6}{5}, \color{blue}{\left(\frac{8}{25} \cdot \frac{1}{t}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{6}{5}, \left(\frac{\frac{8}{25} \cdot 1}{\color{blue}{t}}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{6}{5}, \left(\frac{\frac{8}{25}}{t}\right)\right)\right) \]

      4. /-lowering-/.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{6}{5}, \mathsf{/.f64}\left(\frac{8}{25}, \color{blue}{t}\right)\right)\right) \]

   9. Simplified 97.9%

      \[\leadsto \frac{1}{\color{blue}{1.2 + \frac{0.32}{t}}} \]

   if -0.849999999999999978 < t < 0.39000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. 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 99.2%

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

   5. Simplified 99.2%

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

Alternative 9: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.8) t_1 (if (<= t 0.55) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.55) {
		tmp = 0.5 + (t * t);
	} 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) / t)
    if (t <= (-0.8d0)) then
        tmp = t_1
    else if (t <= 0.55d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.55) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.8:
		tmp = t_1
	elif t <= 0.55:
		tmp = 0.5 + (t * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.8)
		tmp = t_1;
	elseif (t <= 0.55)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.8)
		tmp = t_1;
	elseif (t <= 0.55)
		tmp = 0.5 + (t * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.8], t$95$1, If[LessEqual[t, 0.55], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.55000000000000004 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. 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 97.8%

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

   5. Simplified 97.8%

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

   if -0.80000000000000004 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. 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 99.2%

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

   5. Simplified 99.2%

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

Alternative 10: 98.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.42)
   0.8333333333333334
   (if (<= t 0.6) (+ 0.5 (* t t)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.6) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.42d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.6d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.42) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.6) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.42:
		tmp = 0.8333333333333334
	elif t <= 0.6:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.6)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.42)
		tmp = 0.8333333333333334;
	elseif (t <= 0.6)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.42], 0.8333333333333334, If[LessEqual[t, 0.6], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.419999999999999984 or 0.599999999999999978 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 96.1%

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

   if -0.419999999999999984 < t < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. 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 99.2%

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

   5. Simplified 99.2%

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

Alternative 11: 98.5% 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%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 96.7%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 98.2%

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

Alternative 12: 59.8% accurate, 51.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around 0

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

5. Simplified 59.6%

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

Reproduce

herbie shell --seed 2024156 
(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))))))))