Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. rgt-mult-inverse N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 100.0

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. rgt-mult-inverse N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{1 + t}, 2 + \frac{-2}{1 + t}, 2\right)} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 - \frac{-1}{t}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{\mathsf{fma}\left(t, -0.037037037037037035, -0.04938271604938271\right)}{t \cdot t}}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (- 1.0 (/ -1.0 t)))) 4e-5)
   (fma t (fma -2.0 (* t t) t) 0.5)
   (-
    1.0
    (+
     (/
      (+
       0.2222222222222222
       (/ (fma t -0.037037037037037035 -0.04938271604938271) (* t t)))
      t)
     0.16666666666666666))))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / (1.0 - (-1.0 / t)))) <= 4e-5) {
		tmp = fma(t, fma(-2.0, (t * t), t), 0.5);
	} else {
		tmp = 1.0 - (((0.2222222222222222 + (fma(t, -0.037037037037037035, -0.04938271604938271) / (t * t))) / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 - Float64(-1.0 / t)))) <= 4e-5)
		tmp = fma(t, fma(-2.0, Float64(t * t), t), 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 + Float64(fma(t, -0.037037037037037035, -0.04938271604938271) / Float64(t * t))) / t) + 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 - N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(t * N[(-2.0 * N[(t * t), $MachinePrecision] + t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(N[(N[(0.2222222222222222 + N[(N[(t * -0.037037037037037035 + -0.04938271604938271), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(-2, \color{blue}{t \cdot t}, t\right), 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)} \]

   if 4.00000000000000033e-5 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. div-sub N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222 + \frac{\mathsf{fma}\left(t, -0.037037037037037035, -0.04938271604938271\right)}{t \cdot t}}{t} + 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{1 - \frac{-1}{t}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(-2, t \cdot t, t\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{\mathsf{fma}\left(t, -0.037037037037037035, -0.04938271604938271\right)}{t \cdot t}}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))