Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 11

Speedup: 1.5×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.74:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (-
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            0.2222222222222222)
           t))))
   (if (<= t -0.48)
     t_1
     (if (<= t 0.74) (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0))))) 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 <= 0.74) {
		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 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    if (t <= (-0.48d0)) then
        tmp = t_1
    else if (t <= 0.74d0) 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 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	double tmp;
	if (t <= -0.48) {
		tmp = t_1;
	} else if (t <= 0.74) {
		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 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	tmp = 0
	if t <= -0.48:
		tmp = t_1
	elif t <= 0.74:
		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(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.48)
		tmp = t_1;
	elseif (t <= 0.74)
		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 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.48)
		tmp = t_1;
	elseif (t <= 0.74)
		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[(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, 0.74], 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 < -0.47999999999999998 or 0.73999999999999999 < 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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 98.8%

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

   7. Simplified 98.8%

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

   if -0.47999999999999998 < t < 0.73999999999999999

   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 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.74:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t))))
   (if (<= t -0.58)
     t_1
     (if (<= t 0.58) (+ 0.5 (* (* t t) (+ 1.0 (* t (+ 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 <= -0.58) {
		tmp = t_1;
	} else if (t <= 0.58) {
		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 + (((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t)
    if (t <= (-0.58d0)) then
        tmp = t_1
    else if (t <= 0.58d0) 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 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.58) {
		tmp = t_1;
	} else if (t <= 0.58) {
		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 + (((0.037037037037037035 / t) + -0.2222222222222222) / t)
	tmp = 0
	if t <= -0.58:
		tmp = t_1
	elif t <= 0.58:
		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(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.58)
		tmp = t_1;
	elseif (t <= 0.58)
		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 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.58)
		tmp = t_1;
	elseif (t <= 0.58)
		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[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.58], t$95$1, If[LessEqual[t, 0.58], 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 < -0.57999999999999996 or 0.57999999999999996 < 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. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   if -0.57999999999999996 < 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 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 100.0%

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

   7. Simplified 100.0%

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

Alternative 4: 99.3% 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. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   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. *-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 99.9%

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

   7. Simplified 99.9%

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

Alternative 5: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.24:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t))))
   (if (<= t -0.82) t_1 (if (<= t 0.24) (+ 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 <= 0.24) {
		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 <= 0.24d0) 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 <= 0.24) {
		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 <= 0.24:
		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 <= 0.24)
		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 <= 0.24)
		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, 0.24], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.23999999999999999 < 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. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   if -0.819999999999999951 < t < 0.23999999999999999

   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 99.7%

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

   7. Simplified 99.7%

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

Alternative 6: 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 7: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.8)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (if (<= t 0.58)
     (+ 0.5 (* t t))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.8) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.8d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.8) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.58) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.8:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 0.58:
		tmp = 0.5 + (t * t)
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.8)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.8)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 0.58)
		tmp = 0.5 + (t * t);
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.8], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.80000000000000004

   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 97.7%

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

   7. Simplified 97.7%

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

   if -0.80000000000000004 < 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 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 99.7%

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

   7. Simplified 99.7%

      \[\leadsto \color{blue}{0.5 + t \cdot t} \]

   if 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 98.5%

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

   5. Simplified 98.5%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.8) t_1 (if (<= t 0.58) (+ 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 <= 0.58) {
		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 <= 0.58d0) 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 <= 0.58) {
		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 <= 0.58:
		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 <= 0.58)
		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 <= 0.58)
		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, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.57999999999999996 < 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 98.0%

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

   7. Simplified 98.0%

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

   if -0.80000000000000004 < 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 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 99.7%

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

   7. Simplified 99.7%

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

Alternative 9: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.58) (+ 0.5 (* t t)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		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 <= 0.58d0) 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 <= 0.58) {
		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 <= 0.58:
		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 <= 0.58)
		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 <= 0.58)
		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, 0.58], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.92000000000000004 or 0.57999999999999996 < 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 96.9%

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

   if -0.92000000000000004 < 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 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 99.7%

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

   7. Simplified 99.7%

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

Alternative 10: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		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 <= (-0.34d0)) 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 <= -0.34) {
		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 <= -0.34:
		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 <= -0.34)
		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 <= -0.34)
		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, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 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 96.9%

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

   if -0.340000000000000024 < 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 99.4%

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

Alternative 11: 59.2% 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 2024139 
(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)))))))))