Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 15.6s

Alternatives: 10

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

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

Fortran

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

Java

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

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

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

Fortran

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

Java

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

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+16}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+15}:\\ \;\;\;\;0.5 + \frac{t \cdot t}{1 + \frac{t}{\frac{1}{2 + t \cdot 3}}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -5e+16)
   0.8333333333333334
   (if (<= t 5e+15)
     (+ 0.5 (/ (* t t) (+ 1.0 (/ t (/ 1.0 (+ 2.0 (* t 3.0)))))))
     0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -5e+16) {
		tmp = 0.8333333333333334;
	} else if (t <= 5e+15) {
		tmp = 0.5 + ((t * t) / (1.0 + (t / (1.0 / (2.0 + (t * 3.0))))));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d+16)) then
        tmp = 0.8333333333333334d0
    else if (t <= 5d+15) then
        tmp = 0.5d0 + ((t * t) / (1.0d0 + (t / (1.0d0 / (2.0d0 + (t * 3.0d0))))))
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -5e+16) {
		tmp = 0.8333333333333334;
	} else if (t <= 5e+15) {
		tmp = 0.5 + ((t * t) / (1.0 + (t / (1.0 / (2.0 + (t * 3.0))))));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -5e+16:
		tmp = 0.8333333333333334
	elif t <= 5e+15:
		tmp = 0.5 + ((t * t) / (1.0 + (t / (1.0 / (2.0 + (t * 3.0))))))
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -5e+16)
		tmp = 0.8333333333333334;
	elseif (t <= 5e+15)
		tmp = Float64(0.5 + Float64(Float64(t * t) / Float64(1.0 + Float64(t / Float64(1.0 / Float64(2.0 + Float64(t * 3.0)))))));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -5e+16)
		tmp = 0.8333333333333334;
	elseif (t <= 5e+15)
		tmp = 0.5 + ((t * t) / (1.0 + (t / (1.0 / (2.0 + (t * 3.0))))));
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -5e+16], 0.8333333333333334, If[LessEqual[t, 5e+15], N[(0.5 + N[(N[(t * t), $MachinePrecision] / N[(1.0 + N[(t / N[(1.0 / N[(2.0 + N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -5e16 or 5e15 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

   7. Simplified 100.0%

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

   if -5e16 < t < 5e15

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

   6. Applied egg-rr 100.0%

      \[\leadsto 0.5 + \color{blue}{\frac{\frac{t \cdot t}{t + 1}}{\left(1 + \frac{\left(t \cdot t\right) \cdot 2}{\left(t + 1\right) \cdot \left(t + 1\right)}\right) \cdot \left(t + 1\right)}} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. times-frac N/A

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

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

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

      13. associate-/l* N/A

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

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

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

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

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. +-lowering-+.f64 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \color{blue}{\left(1 + t \cdot \left(2 + 3 \cdot t\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. lft-mult-inverse N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(1 + t \cdot \left(2 \cdot \left(\frac{1}{t} \cdot t\right) + 3 \cdot t\right)\right)\right)\right) \]

       3. associate-*l* N/A

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

       4. distribute-rgt-in N/A

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

       5. +-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. +-commutative N/A

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

       15. distribute-rgt-in N/A

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

       16. associate-*l* N/A

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

       17. lft-mult-inverse N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(2 \cdot 1 + 3 \cdot t\right)\right)\right)\right)\right) \]

       18. metadata-eval N/A

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

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

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

       20. *-commutative N/A

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

       21. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

       \[\leadsto 0.5 + \frac{t \cdot t}{\color{blue}{1 + t \cdot \left(2 + t \cdot 3\right)}} \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. clear-num N/A

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

       6. flip-+ N/A

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

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

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

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

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

       9. *-lowering-*.f64 100.0%

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

   13. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

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

Fortran

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

Java

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

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 3: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+16}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+15}:\\ \;\;\;\;0.5 + \frac{t \cdot t}{1 + t \cdot \left(2 + t \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -5e+16)
   0.8333333333333334
   (if (<= t 5e+15)
     (+ 0.5 (/ (* t t) (+ 1.0 (* t (+ 2.0 (* t 3.0))))))
     0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -5e+16) {
		tmp = 0.8333333333333334;
	} else if (t <= 5e+15) {
		tmp = 0.5 + ((t * t) / (1.0 + (t * (2.0 + (t * 3.0)))));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d+16)) then
        tmp = 0.8333333333333334d0
    else if (t <= 5d+15) then
        tmp = 0.5d0 + ((t * t) / (1.0d0 + (t * (2.0d0 + (t * 3.0d0)))))
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -5e+16) {
		tmp = 0.8333333333333334;
	} else if (t <= 5e+15) {
		tmp = 0.5 + ((t * t) / (1.0 + (t * (2.0 + (t * 3.0)))));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -5e+16:
		tmp = 0.8333333333333334
	elif t <= 5e+15:
		tmp = 0.5 + ((t * t) / (1.0 + (t * (2.0 + (t * 3.0)))))
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -5e+16)
		tmp = 0.8333333333333334;
	elseif (t <= 5e+15)
		tmp = Float64(0.5 + Float64(Float64(t * t) / Float64(1.0 + Float64(t * Float64(2.0 + Float64(t * 3.0))))));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -5e+16)
		tmp = 0.8333333333333334;
	elseif (t <= 5e+15)
		tmp = 0.5 + ((t * t) / (1.0 + (t * (2.0 + (t * 3.0)))));
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -5e+16], 0.8333333333333334, If[LessEqual[t, 5e+15], N[(0.5 + N[(N[(t * t), $MachinePrecision] / N[(1.0 + N[(t * N[(2.0 + N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -5e16 or 5e15 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

   7. Simplified 100.0%

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

   if -5e16 < t < 5e15

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

   6. Applied egg-rr 100.0%

      \[\leadsto 0.5 + \color{blue}{\frac{\frac{t \cdot t}{t + 1}}{\left(1 + \frac{\left(t \cdot t\right) \cdot 2}{\left(t + 1\right) \cdot \left(t + 1\right)}\right) \cdot \left(t + 1\right)}} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. times-frac N/A

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

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

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

      13. associate-/l* N/A

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

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

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

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

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. +-lowering-+.f64 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \color{blue}{\left(1 + t \cdot \left(2 + 3 \cdot t\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. lft-mult-inverse N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(1 + t \cdot \left(2 \cdot \left(\frac{1}{t} \cdot t\right) + 3 \cdot t\right)\right)\right)\right) \]

       3. associate-*l* N/A

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

       4. distribute-rgt-in N/A

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

       5. +-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. +-commutative N/A

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

       15. distribute-rgt-in N/A

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

       16. associate-*l* N/A

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

       17. lft-mult-inverse N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(2 \cdot 1 + 3 \cdot t\right)\right)\right)\right)\right) \]

       18. metadata-eval N/A

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

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

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

       20. *-commutative N/A

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

       21. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

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

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (+
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            -0.2222222222222222)
           t))))
   (if (<= t -0.5)
     t_1
     (if (<= t 0.56) (+ 0.5 (* (* t t) (+ 1.0 (* 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 <= -0.5) {
		tmp = t_1;
	} else if (t <= 0.56) {
		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 + (0.04938271604938271d0 / t)) / t) + (-0.2222222222222222d0)) / t)
    if (t <= (-0.5d0)) then
        tmp = t_1
    else if (t <= 0.56d0) 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 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.5) {
		tmp = t_1;
	} else if (t <= 0.56) {
		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 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t)
	tmp = 0
	if t <= -0.5:
		tmp = t_1
	elif t <= 0.56:
		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(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.5)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(Float64(t * 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 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.5)
		tmp = t_1;
	elseif (t <= 0.56)
		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[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.5], t$95$1, If[LessEqual[t, 0.56], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.5 or 0.56000000000000005 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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. +-commutative N/A

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

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

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

   7. Simplified 99.1%

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

   if -0.5 < t < 0.56000000000000005

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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. +-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.6)
     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 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / 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 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / 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 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / 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 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / 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(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.6)
		tmp = t_1;
	elseif (t <= 0.45)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * -2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.6)
		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[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.6], t$95$1, If[LessEqual[t, 0.45], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.450000000000000011 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. remove-double-neg N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-in N/A

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

      11. sub-neg N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. div-sub 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

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

   7. Simplified 98.7%

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

   if -0.599999999999999978 < t < 0.450000000000000011

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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. +-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.34:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.82) t_1 (if (<= t 0.34) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.34) {
		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) + (0.037037037037037035d0 / t)) / t)
    if (t <= (-0.82d0)) then
        tmp = t_1
    else if (t <= 0.34d0) 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 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.34) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.82:
		tmp = t_1
	elif t <= 0.34:
		tmp = 0.5 + (t * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.34)
		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 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.34)
		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[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.82], t$95$1, If[LessEqual[t, 0.34], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.340000000000000024 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. remove-double-neg N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. distribute-neg-in N/A

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

      11. sub-neg N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. div-sub 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

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

   7. Simplified 98.7%

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

   if -0.819999999999999951 < t < 0.340000000000000024

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.34:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.8) t_1 (if (<= t 0.56) (+ 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 <= 0.56) {
		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 <= 0.56d0) 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 <= 0.56) {
		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 <= 0.56:
		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 <= 0.56)
		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 <= 0.56)
		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, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.56000000000000005 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 97.8%

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

   7. Simplified 97.8%

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

   if -0.80000000000000004 < t < 0.56000000000000005

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.48) t_1 (if (<= t 0.66) 0.5 t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.48) {
		tmp = t_1;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.48d0)) then
        tmp = t_1
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.48) {
		tmp = t_1;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.48:
		tmp = t_1
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.48)
		tmp = t_1;
	elseif (t <= 0.66)
		tmp = 0.5;
	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 <= -0.48)
		tmp = t_1;
	elseif (t <= 0.66)
		tmp = 0.5;
	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, -0.48], t$95$1, If[LessEqual[t, 0.66], 0.5, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.660000000000000031 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   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 97.8%

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

   7. Simplified 97.8%

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

   if -0.47999999999999998 < t < 0.660000000000000031

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Simplified 99.7%

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

Alternative 9: 98.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

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

   7. Simplified 95.8%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Simplified 99.7%

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

Alternative 10: 58.3% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

3. Simplified 78.1%

   \[\leadsto \color{blue}{0.5 + \frac{\frac{t \cdot t}{\left(1 + t\right) \cdot \left(1 + t\right)}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0

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

7. Simplified 59.1%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))