Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ 1.0 (+ -6.0 (* (/ 2.0 (+ 1.0 t)) (+ (/ 2.0 (- -1.0 t)) 4.0))))))

C

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

Fortran

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

Java

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

Python

def code(t):
	return 1.0 + (1.0 / (-6.0 + ((2.0 / (1.0 + t)) * ((2.0 / (-1.0 - t)) + 4.0))))

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(-6.0 + N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t}}{t}\\ \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.72:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (-
          0.8333333333333334
          (/
           (+
            0.2222222222222222
            (/ (+ -0.037037037037037035 (/ -0.04938271604938271 t)) t))
           t))))
   (if (<= t -0.35)
     t_1
     (if (<= t 0.72) (+ 0.5 (* (* t t) (+ 1.0 (* t (+ 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 <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.72) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (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 <= (-0.35d0)) then
        tmp = t_1
    else if (t <= 0.72d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (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 <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.72) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (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 <= -0.35:
		tmp = t_1
	elif t <= 0.72:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (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 <= -0.35)
		tmp = t_1;
	elseif (t <= 0.72)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * 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 <= -0.35)
		tmp = t_1;
	elseif (t <= 0.72)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (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, -0.35], t$95$1, If[LessEqual[t, 0.72], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.34999999999999998 or 0.71999999999999997 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

   7. Simplified 99.9%

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

   if -0.34999999999999998 < t < 0.71999999999999997

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 99.4%

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

   7. Simplified 99.4%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;1 + \left(-0.16666666666666666 + t\_1\right)\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)))
   (if (<= t -0.58)
     (+ 1.0 (+ -0.16666666666666666 t_1))
     (if (<= t 0.6)
       (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0)))))
       (+ 0.8333333333333334 t_1)))))

C

double code(double t) {
	double t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t;
	double tmp;
	if (t <= -0.58) {
		tmp = 1.0 + (-0.16666666666666666 + t_1);
	} else if (t <= 0.6) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = 0.8333333333333334 + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t
    if (t <= (-0.58d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + t_1)
    else if (t <= 0.6d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (t + (-2.0d0)))))
    else
        tmp = 0.8333333333333334d0 + t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t;
	double tmp;
	if (t <= -0.58) {
		tmp = 1.0 + (-0.16666666666666666 + t_1);
	} else if (t <= 0.6) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = 0.8333333333333334 + t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t
	tmp = 0
	if t <= -0.58:
		tmp = 1.0 + (-0.16666666666666666 + t_1)
	elif t <= 0.6:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))))
	else:
		tmp = 0.8333333333333334 + t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t)
	tmp = 0.0
	if (t <= -0.58)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + t_1));
	elseif (t <= 0.6)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * Float64(t + -2.0)))));
	else
		tmp = Float64(0.8333333333333334 + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t;
	tmp = 0.0;
	if (t <= -0.58)
		tmp = 1.0 + (-0.16666666666666666 + t_1);
	elseif (t <= 0.6)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	else
		tmp = 0.8333333333333334 + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -0.58], N[(1.0 + N[(-0.16666666666666666 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.6], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.57999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r- N/A

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

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

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. div-sub N/A

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

      18. neg-sub0 N/A

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

      19. distribute-neg-frac N/A

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

      20. mul-1-neg N/A

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

   7. Simplified 99.7%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)} \]

   if -0.57999999999999996 < t < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 99.4%

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

   7. Simplified 99.4%

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

   if 0.599999999999999978 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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. mul-1-neg N/A

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

   7. Simplified 99.7%

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

Alternative 4: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;1 + \left(-0.16666666666666666 + t\_1\right)\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t)))
   (if (<= t -0.6)
     (+ 1.0 (+ -0.16666666666666666 t_1))
     (if (<= t 0.45)
       (+ 0.5 (* t (* t (+ 1.0 (* t -2.0)))))
       (+ 0.8333333333333334 t_1)))))

C

double code(double t) {
	double t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t;
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-0.16666666666666666 + t_1);
	} else if (t <= 0.45) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = 0.8333333333333334 + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t
    if (t <= (-0.6d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + t_1)
    else if (t <= 0.45d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = 0.8333333333333334d0 + t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t;
	double tmp;
	if (t <= -0.6) {
		tmp = 1.0 + (-0.16666666666666666 + t_1);
	} else if (t <= 0.45) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = 0.8333333333333334 + t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t
	tmp = 0
	if t <= -0.6:
		tmp = 1.0 + (-0.16666666666666666 + t_1)
	elif t <= 0.45:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = 0.8333333333333334 + t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + t_1));
	elseif (t <= 0.45)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = Float64(0.8333333333333334 + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = ((0.037037037037037035 / t) + -0.2222222222222222) / t;
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 1.0 + (-0.16666666666666666 + t_1);
	elseif (t <= 0.45)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = 0.8333333333333334 + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -0.6], N[(1.0 + N[(-0.16666666666666666 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.45], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r- N/A

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

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

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. div-sub N/A

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

      18. neg-sub0 N/A

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

      19. distribute-neg-frac N/A

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

      20. mul-1-neg N/A

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

   7. Simplified 99.7%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\right)} \]

   if -0.599999999999999978 < t < 0.450000000000000011

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   if 0.450000000000000011 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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. mul-1-neg N/A

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

   7. Simplified 99.7%

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

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t))))
   (if (<= t -0.6)
     t_1
     (if (<= t 0.45) (+ 0.5 (* t (* t (+ 1.0 (* t -2.0))))) 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 * (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 + (((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t)
    if (t <= (-0.6d0)) then
        tmp = t_1
    else if (t <= 0.45d0) 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 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1;
	} else if (t <= 0.45) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t)
	tmp = 0
	if t <= -0.6:
		tmp = t_1
	elif t <= 0.45:
		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(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(1.0 + Float64(t * -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.6)
		tmp = t_1;
	elseif (t <= 0.45)
		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[(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[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.450000000000000011 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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. mul-1-neg N/A

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

   7. Simplified 99.7%

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

   if -0.599999999999999978 < t < 0.450000000000000011

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

Alternative 6: 99.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t))))
   (if (<= t -0.82) t_1 (if (<= t 0.235) (+ 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 <= 0.235) {
		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 <= 0.235d0) 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 <= 0.235) {
		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 <= 0.235:
		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 <= 0.235)
		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 <= 0.235)
		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, 0.235], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.23499999999999999 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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. mul-1-neg N/A

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

   7. Simplified 99.1%

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

   if -0.819999999999999951 < t < 0.23499999999999999

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

Alternative 7: 99.3% accurate, 1.9× 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%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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 99.3%

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

   7. Simplified 99.3%

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

   if -0.80000000000000004 < t < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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 99.3%

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

   7. Simplified 99.3%

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

Alternative 8: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.6) (+ 0.5 (* t t)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		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 <= (-0.92d0)) 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 <= -0.92) {
		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 <= -0.92:
		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 <= -0.92)
		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 <= -0.92)
		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, -0.92], 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 < -0.92000000000000004 or 0.599999999999999978 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 97.9%

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

   if -0.92000000000000004 < t < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   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 99.3%

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

   7. Simplified 99.3%

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

Alternative 9: 98.7% accurate, 2.6× 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%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 97.9%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 98.8%

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

Alternative 10: 58.4% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around 0

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

7. Simplified 58.6%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))