Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 9.3s

Precision: binary64

Cost: 1728

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{4}{1 + t} + -8}{1 + t}\\ \frac{t_1 + 5}{t_1 + 6} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   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
 (let* ((t_1 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t))))
   (/ (+ t_1 5.0) (+ t_1 6.0))))

C

double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (t_1 + 5.0) / (t_1 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))))) / (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
    real(8) :: t_1
    t_1 = ((4.0d0 / (1.0d0 + t)) + (-8.0d0)) / (1.0d0 + t)
    code = (t_1 + 5.0d0) / (t_1 + 6.0d0)
end function

Java

public static double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (t_1 + 5.0) / (t_1 + 6.0);
}

Python

def code(t):
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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):
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)
	return (t_1 + 5.0) / (t_1 + 6.0)

Julia

function code(t)
	return Float64(Float64(1.0 + 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)))))) / 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)
	t_1 = Float64(Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0) / Float64(1.0 + t))
	return Float64(Float64(t_1 + 5.0) / Float64(t_1 + 6.0))
end

MATLAB

function tmp = code(t)
	tmp = (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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)
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	tmp = (t_1 + 5.0) / (t_1 + 6.0);
end

Wolfram

code[t_] := N[(N[(1.0 + 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] / 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]

↓

code[t_] := Block[{t$95$1 = N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 + 5.0), $MachinePrecision] / N[(t$95$1 + 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}} \]
   Proof
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 2)) (+.f64 1 t)) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (+.f64 (Rewrite<= rgt-mult-inverse_binary64 (*.f64 t (/.f64 1 t))) t)) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (+.f64 (*.f64 t (/.f64 1 t)) (Rewrite<= *-rgt-identity_binary64 (*.f64 t 1)))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (Rewrite<= distribute-lft-in_binary64 (*.f64 t (+.f64 (/.f64 1 t) 1)))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (*.f64 t (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 1 t))))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 (/.f64 1 t)) t))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (*.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (neg.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) -8) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (Rewrite<= metadata-eval (*.f64 -2 4))) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) 4)) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 2) (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 (neg.f64 2) (+.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (*.f64 2 2)))) (+.f64 1 t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (+.f64 (Rewrite<= rgt-mult-inverse_binary64 (*.f64 t (/.f64 1 t))) t)) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (+.f64 (*.f64 t (/.f64 1 t)) (Rewrite<= *-rgt-identity_binary64 (*.f64 t 1)))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (Rewrite<= distribute-lft-in_binary64 (*.f64 t (+.f64 (/.f64 1 t) 1)))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 64 points increase in error, 0 points decrease in error
   (/.f64 (+.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)))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 24 points increase in error, 52 points decrease in error
   (/.f64 (+.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))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.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)))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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))))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.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))))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.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))))))) 5) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (Rewrite<= metadata-eval (+.f64 1 4))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 1 (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (+.f64 1 (*.f64 2 2))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 1 (*.f64 2 2)) (+.f64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 1 (*.f64 2 2)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 1 (*.f64 2 2)) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (*.f64 2 2) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 1 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 1 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (*.f64 2 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 5 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (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)))))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 5 points decrease in error
   (/.f64 (+.f64 1 (*.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))))))) (+.f64 (/.f64 (+.f64 (/.f64 4 (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 2)) (+.f64 1 t)) -8) (+.f64 1 t)) 6)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (+.f64 (Rewrite<= rgt-mult-inverse_binary64 (*.f64 t (/.f64 1 t))) t)) -8) (+.f64 1 t)) 6)): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (+.f64 (*.f64 t (/.f64 1 t)) (Rewrite<= *-rgt-identity_binary64 (*.f64 t 1)))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (Rewrite<= distribute-lft-in_binary64 (*.f64 t (+.f64 (/.f64 1 t) 1)))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (/.f64 (*.f64 2 2) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 (/.f64 1 t)) t))) -8) (+.f64 1 t)) 6)): 54 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 54 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (*.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (neg.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) -8) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (Rewrite<= metadata-eval (*.f64 -2 4))) (+.f64 1 t)) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) 4)) (+.f64 1 t)) 6)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (+.f64 (*.f64 (neg.f64 2) (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 2) (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 1 t)) 6)): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (Rewrite<= distribute-lft-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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 24 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 24 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 24 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (+.f64 (Rewrite<= rgt-mult-inverse_binary64 (*.f64 t (/.f64 1 t))) t)) 6)): 5 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (+.f64 (*.f64 t (/.f64 1 t)) (Rewrite<= *-rgt-identity_binary64 (*.f64 t 1)))) 6)): 0 points increase in error, 5 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (/.f64 (*.f64 2 (+.f64 (neg.f64 2) (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) -1))) (Rewrite<= distribute-lft-in_binary64 (*.f64 t (+.f64 (/.f64 1 t) 1)))) 6)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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)): 54 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 54 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.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))))))) 6)): 24 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (Rewrite<= metadata-eval (+.f64 2 4)))): 0 points increase in error, 24 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (+.f64 (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (+.f64 2 (*.f64 2 2)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 2 (*.f64 2 2)) (*.f64 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))))): 57 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (+.f64 2 (*.f64 2 2))) (*.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, 57 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 2 (*.f64 2 2)) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (*.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 2 (+.f64 (*.f64 2 2) (*.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (*.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (+.f64 2 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 2 (neg.f64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))) (*.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 (+.f64 2 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (*.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (Rewrite<= associate-+r+_binary64 (+.f64 2 (+.f64 (*.f64 2 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (*.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.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

3. Final simplification 0.0

   \[\leadsto \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6} \]

Alternatives

Alternative 1
Error	0.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 1.65:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	836

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + t_1\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 4
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 5
Error	0.9
Cost	328

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

Alternative 6
Error	26.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))