Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 7.7s

Precision: binary64

Cost: 1088

Math

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

↓

\[1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6} \]

FPCore

(FPCore (t)
 :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)))))))))

↓

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t)) 6.0))))

C

double code(double t) {
	return 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)))))));
}

↓

double code(double t) {
	return 1.0 + (-1.0 / ((((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)) + 6.0));
}

Fortran

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

↓

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + ((-1.0d0) / ((((4.0d0 / (1.0d0 + t)) + (-8.0d0)) / (1.0d0 + t)) + 6.0d0))
end function

Java

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

↓

public static double code(double t) {
	return 1.0 + (-1.0 / ((((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)) + 6.0));
}

Python

def code(t):
	return 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)))))))

↓

def code(t):
	return 1.0 + (-1.0 / ((((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)) + 6.0))

Julia

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

↓

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(Float64(Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0) / Float64(1.0 + t)) + 6.0)))
end

MATLAB

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

↓

function tmp = code(t)
	tmp = 1.0 + (-1.0 / ((((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)) + 6.0));
end

Wolfram

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

↓

code[t_] := N[(1.0 + N[(-1.0 / N[(N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}} \]
   Proof
   (+.f64 1 (/.f64 -1 (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 -2 -2)) (+.f64 1 t)) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) -2) (+.f64 1 t)) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (neg.f64 2) (Rewrite<= metadata-eval (neg.f64 2))) (+.f64 1 t)) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (neg.f64 2) (neg.f64 2)) (+.f64 (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 t) t)) t)) -8) (+.f64 1 t)) 6))): 15 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (neg.f64 2) (neg.f64 2)) (+.f64 (*.f64 (/.f64 1 t) t) (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)))) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (neg.f64 2) (neg.f64 2)) (Rewrite=> distribute-rgt-out_binary64 (*.f64 t (+.f64 (/.f64 1 t) 1)))) -8) (+.f64 1 t)) 6))): 1 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (neg.f64 2) (neg.f64 2)) (*.f64 t (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 1 t))))) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 (neg.f64 2) (neg.f64 2)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 (/.f64 1 t)) t))) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (neg.f64 2) (*.f64 (+.f64 1 (/.f64 1 t)) t)) (neg.f64 2))) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))) (neg.f64 2)) -8) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (neg.f64 2)) -8) (+.f64 1 t)) 6))): 1 points increase in error, 15 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 2)) (Rewrite<= metadata-eval (*.f64 4 -2))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 2)) (*.f64 (Rewrite<= metadata-eval (*.f64 2 2)) -2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 2)) (*.f64 (*.f64 2 2) (Rewrite<= metadata-eval (neg.f64 2)))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 (neg.f64 2) (+.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (*.f64 2 2)))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 (neg.f64 2) (+.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (Rewrite=> metadata-eval 4))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 (neg.f64 2) (+.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (Rewrite<= metadata-eval (+.f64 2 2)))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 (neg.f64 2) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) 2) 2))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 (neg.f64 2) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) 2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 (neg.f64 2) (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) 2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (neg.f64 2) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 2) 2))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 2))) (*.f64 (neg.f64 2) 2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (Rewrite=> metadata-eval -2)) (*.f64 (neg.f64 2) 2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (Rewrite<= metadata-eval (*.f64 -1 2))) (*.f64 (neg.f64 2) 2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1) 2)) (*.f64 (neg.f64 2) 2)) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 2 (+.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1) (neg.f64 2)))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 2 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1)))) (+.f64 1 t)) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (+.f64 (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 t) t)) t)) 6))): 14 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (+.f64 (*.f64 (/.f64 1 t) t) (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (Rewrite=> distribute-rgt-out_binary64 (*.f64 t (+.f64 (/.f64 1 t) 1)))) 6))): 2 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (*.f64 t (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 1 t))))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 (/.f64 1 t)) t))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)) (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1)))) 6))): 0 points increase in error, 1 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) 6))): 0 points increase in error, 15 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (neg.f64 2) (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (*.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1) (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) 6))): 0 points increase in error, 1 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (*.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1) (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (+.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (*.f64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1) (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (Rewrite<= associate-*r*_binary64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (*.f64 -1 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) 6))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (Rewrite<= metadata-eval (+.f64 2 4))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 2 (Rewrite<= metadata-eval (*.f64 2 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 2 (*.f64 2 2)) (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (Rewrite<= associate-+r+_binary64 (+.f64 2 (+.f64 (*.f64 2 2) (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 2) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 2 (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 2 (+.f64 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 2 (+.f64 (*.f64 2 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (Rewrite=> *-commutative_binary64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 2 (Rewrite<= distribute-rgt-in_binary64 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (+.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (neg.f64 1) (+.f64 2 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (Rewrite<= sub-neg_binary64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 (+.f64 2 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 1 (+.f64 2 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

   \[\leadsto 1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6} \]

Alternatives

Alternative 1
Error	0.7
Cost	1224

\[\begin{array}{l} \mathbf{if}\;t \leq -1189882512.6421301:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.0007913128522455004:\\ \;\;\;\;1 + \frac{-1}{2 + \left(t \cdot t\right) \cdot \left(4 + t \cdot -8\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -1189882512.6421301:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.0007913128522455004:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(4 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1189882512.6421301:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.0007913128522455004:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 4
Error	0.8
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -1189882512.6421301:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.0007913128522455004:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1189882512.6421301:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.0007913128522455004:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -1189882512.6421301:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.0007913128522455004:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 7
Error	26.4
Cost	64

\[0.8333333333333334 \]

Reproduce

herbie shell --seed 2022316 
(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)))))))))