Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 12.8s

Alternatives: 10

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 * t) / (t + 1.0);
	double t_2 = t_1 * t_1;
	return (t_2 + 1.0) / (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) / (t + 1.0d0)
    t_2 = t_1 * t_1
    code = (t_2 + 1.0d0) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(t$95$2 + 1.0), $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. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot \left(t \cdot 4\right)}{\left(t + 1\right) \cdot \left(t + 1\right)}\\ \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 1.999999999:\\ \;\;\;\;\frac{t\_1 + 1}{2 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t (* t 4.0)) (* (+ t 1.0) (+ t 1.0)))))
   (if (<= (/ (* 2.0 t) (+ t 1.0)) 1.999999999)
     (/ (+ t_1 1.0) (+ 2.0 t_1))
     (+
      0.8333333333333334
      (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t))))))

C

double code(double t) {
	double t_1 = (t * (t * 4.0)) / ((t + 1.0) * (t + 1.0));
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1.999999999) {
		tmp = (t_1 + 1.0) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(t * Float64(t * 4.0)) / Float64(Float64(t + 1.0) * Float64(t + 1.0)))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1.999999999)
		tmp = Float64(Float64(t_1 + 1.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t + 1.0), $MachinePrecision] * N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1.999999999], N[(N[(t$95$1 + 1.0), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.9999999989999999

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

   4. Applied rewrites 100.0%

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

   if 1.9999999989999999 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. unsub-neg N/A

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

      13. mul-1-neg N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto 0.8333333333333334 + \color{blue}{\frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 1.999999999:\\ \;\;\;\;\frac{\frac{t \cdot \left(t \cdot 4\right)}{\left(t + 1\right) \cdot \left(t + 1\right)} + 1}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(t + 1\right) \cdot \left(t + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}\\ \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 1.9999999999519582:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot 4, t\_1, 1\right)}{\mathsf{fma}\left(t \cdot 4, t\_1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (fma t (+ 2.0 t) 1.0))))
   (if (<= (/ (* 2.0 t) (+ t 1.0)) 1.9999999999519582)
     (/ (fma (* t 4.0) t_1 1.0) (fma (* t 4.0) t_1 2.0))
     (+
      0.8333333333333334
      (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t))))))

C

double code(double t) {
	double t_1 = t / fma(t, (2.0 + t), 1.0);
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1.9999999999519582) {
		tmp = fma((t * 4.0), t_1, 1.0) / fma((t * 4.0), t_1, 2.0);
	} else {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(t / fma(t, Float64(2.0 + t), 1.0))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1.9999999999519582)
		tmp = Float64(fma(Float64(t * 4.0), t_1, 1.0) / fma(Float64(t * 4.0), t_1, 2.0));
	else
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(t * N[(2.0 + t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1.9999999999519582], N[(N[(N[(t * 4.0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] / N[(N[(t * 4.0), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.99999999995195821

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)}} \]

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{t \cdot \left(2 + t\right) + 1}}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{2 \cdot 1} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      3. lft-mult-inverse N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(2 \cdot \color{blue}{\left(\frac{1}{t} \cdot t\right)} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot t} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{t \cdot \left(2 \cdot \frac{1}{t}\right)} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(t \cdot \left(2 \cdot \frac{1}{t}\right) + \color{blue}{t \cdot 1}\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot \frac{1}{t} + 1\right)\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(t \cdot \color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right)}\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(t \cdot t\right) \cdot \left(1 + 2 \cdot \frac{1}{t}\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{{t}^{2}} \cdot \left(1 + 2 \cdot \frac{1}{t}\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right) \cdot {t}^{2}} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + 2 \cdot \frac{1}{t}\right) \cdot \color{blue}{\left(t \cdot t\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right) \cdot t} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{t \cdot \left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\mathsf{fma}\left(t, \left(1 + 2 \cdot \frac{1}{t}\right) \cdot t, 1\right)}}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\mathsf{fma}\left(t, 2 + t, 1\right)}}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{t \cdot \left(2 + t\right) + 1}}, 2\right)} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{2 \cdot 1} + t\right) + 1}, 2\right)} \]

       3. lft-mult-inverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(2 \cdot \color{blue}{\left(\frac{1}{t} \cdot t\right)} + t\right) + 1}, 2\right)} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot t} + t\right) + 1}, 2\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{t \cdot \left(2 \cdot \frac{1}{t}\right)} + t\right) + 1}, 2\right)} \]

       6. *-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(t \cdot \left(2 \cdot \frac{1}{t}\right) + \color{blue}{t \cdot 1}\right) + 1}, 2\right)} \]

       7. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot \frac{1}{t} + 1\right)\right)} + 1}, 2\right)} \]

       8. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(t \cdot \color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right)}\right) + 1}, 2\right)} \]

       9. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(t \cdot t\right) \cdot \left(1 + 2 \cdot \frac{1}{t}\right)} + 1}, 2\right)} \]

       10. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{{t}^{2}} \cdot \left(1 + 2 \cdot \frac{1}{t}\right) + 1}, 2\right)} \]

       11. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right) \cdot {t}^{2}} + 1}, 2\right)} \]

       12. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + 2 \cdot \frac{1}{t}\right) \cdot \color{blue}{\left(t \cdot t\right)} + 1}, 2\right)} \]

       13. associate-*r* N/A

           \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right) \cdot t} + 1}, 2\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{t \cdot \left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right)} + 1}, 2\right)} \]

       15. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\mathsf{fma}\left(t, \left(1 + 2 \cdot \frac{1}{t}\right) \cdot t, 1\right)}}, 2\right)} \]

   12. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\mathsf{fma}\left(t, 2 + t, 1\right)}}, 2\right)} \]

   if 1.99999999995195821 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. unsub-neg N/A

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

      13. mul-1-neg N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto 0.8333333333333334 + \color{blue}{\frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 1.9999999999519582:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.04938271604938271}{t \cdot \left(t \cdot t\right)} + \left(\frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t} - -0.8333333333333334\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ t 1.0)) 1e-5)
   (fma t t 0.5)
   (+
    (/ 0.04938271604938271 (* t (* t t)))
    (-
     (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t))
     -0.8333333333333334))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1e-5) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = (0.04938271604938271 / (t * (t * t))) + ((fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t)) - -0.8333333333333334);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1e-5)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(Float64(0.04938271604938271 / Float64(t * Float64(t * t))) + Float64(Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)) - -0.8333333333333334));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1e-5], N[(t * t + 0.5), $MachinePrecision], N[(N[(0.04938271604938271 / N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 1.00000000000000008e-5 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)}} \]

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{t \cdot \left(2 + t\right) + 1}}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{2 \cdot 1} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      3. lft-mult-inverse N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(2 \cdot \color{blue}{\left(\frac{1}{t} \cdot t\right)} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot t} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(\color{blue}{t \cdot \left(2 \cdot \frac{1}{t}\right)} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(t \cdot \left(2 \cdot \frac{1}{t}\right) + \color{blue}{t \cdot 1}\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot \frac{1}{t} + 1\right)\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{t \cdot \left(t \cdot \color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right)}\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(t \cdot t\right) \cdot \left(1 + 2 \cdot \frac{1}{t}\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{{t}^{2}} \cdot \left(1 + 2 \cdot \frac{1}{t}\right) + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right) \cdot {t}^{2}} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + 2 \cdot \frac{1}{t}\right) \cdot \color{blue}{\left(t \cdot t\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right) \cdot t} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{t \cdot \left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right)} + 1}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\mathsf{fma}\left(t, \left(1 + 2 \cdot \frac{1}{t}\right) \cdot t, 1\right)}}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

   9. Applied rewrites 60.0%

      \[\leadsto \frac{\mathsf{fma}\left(t \cdot 4, \frac{t}{\color{blue}{\mathsf{fma}\left(t, 2 + t, 1\right)}}, 1\right)}{\mathsf{fma}\left(t \cdot 4, \frac{t}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)} \]

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\frac{0.04938271604938271}{t \cdot \left(t \cdot t\right)} - \left(-0.8333333333333334 - \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.04938271604938271}{t \cdot \left(t \cdot t\right)} + \left(\frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t} - -0.8333333333333334\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ t 1.0)) 1e-5)
   (fma t t 0.5)
   (+
    0.8333333333333334
    (/
     (+
      -0.2222222222222222
      (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
     t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1e-5) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334 + ((-0.2222222222222222 + (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1e-5)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1e-5], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 1.00000000000000008e-5 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222 - \frac{-0.037037037037037035 - \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ t 1.0)) 1e-5)
   (fma t t 0.5)
   (+
    0.8333333333333334
    (/ (fma t -0.2222222222222222 0.037037037037037035) (* t t)))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1e-5) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334 + (fma(t, -0.2222222222222222, 0.037037037037037035) / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1e-5)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(fma(t, -0.2222222222222222, 0.037037037037037035) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1e-5], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(N[(t * -0.2222222222222222 + 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 1.00000000000000008e-5 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. unsub-neg N/A

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

      13. mul-1-neg N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.6

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

   8. Applied rewrites 99.6%

      \[\leadsto 0.8333333333333334 + \color{blue}{\frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\mathsf{fma}\left(t, -0.2222222222222222, 0.037037037037037035\right)}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ t 1.0)) 1e-5)
   (fma t t 0.5)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1e-5) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1e-5)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1e-5], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 1.00000000000000008e-5 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ t 1.0)) 1e-5) (fma t t 0.5) 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 1e-5) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1e-5)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1e-5], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.00000000000000008e-5

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 1.00000000000000008e-5 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* 2.0 t) (+ t 1.0)) 1.0) 0.5 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if (((2.0 * t) / (t + 1.0)) <= 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 (((2.0d0 * t) / (t + 1.0d0)) <= 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 (((2.0 * t) / (t + 1.0)) <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 * t) / (t + 1.0)) <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(t + 1.0)) <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 * t) / (t + 1.0)) <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], 1.0], 0.5, 0.8333333333333334]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{0.5} \]

   if 1 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot t}{t + 1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 104.0× 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%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 56.3%

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

Reproduce

herbie shell --seed 2024219 
(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))))))