Octave 3.8, jcobi/4

Average Error: 54.2 → 8.7

Time: 20.0s

Precision: binary64

Cost: 49028

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) \cdot 2\\ t_1 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \left(\beta + \alpha\right) \cdot \mathsf{fma}\left(0.0625, t_0, \left(\beta + \alpha\right) \cdot -0.125\right), \mathsf{fma}\left(4, t_1 + -1, t_1 \cdot 20\right) \cdot -0.00390625\right) + 0.0625 \cdot \left(t_1 + \beta \cdot \alpha\right)}{i \cdot i} + \left(0.0625 + \mathsf{fma}\left(0.0625, \frac{t_0}{i}, -0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\alpha + i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
   (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))))
  (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ beta alpha) 2.0)) (t_1 (pow (+ beta alpha) 2.0)))
   (if (<= beta 4.907639238008554e+137)
     (+
      (/
       (+
        (fma
         -2.0
         (* (+ beta alpha) (fma 0.0625 t_0 (* (+ beta alpha) -0.125)))
         (* (fma 4.0 (+ t_1 -1.0) (* t_1 20.0)) -0.00390625))
        (* 0.0625 (+ t_1 (* beta alpha))))
       (* i i))
      (+ 0.0625 (fma 0.0625 (/ t_0 i) (* -0.125 (/ (+ beta alpha) i)))))
     (*
      (/ i (+ (fma i 2.0 (+ beta alpha)) 1.0))
      (/ (+ alpha i) (+ (+ beta alpha) (fma i 2.0 -1.0)))))))

C

double code(double alpha, double beta, double i) {
	return (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0);
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) * 2.0;
	double t_1 = pow((beta + alpha), 2.0);
	double tmp;
	if (beta <= 4.907639238008554e+137) {
		tmp = ((fma(-2.0, ((beta + alpha) * fma(0.0625, t_0, ((beta + alpha) * -0.125))), (fma(4.0, (t_1 + -1.0), (t_1 * 20.0)) * -0.00390625)) + (0.0625 * (t_1 + (beta * alpha)))) / (i * i)) + (0.0625 + fma(0.0625, (t_0 / i), (-0.125 * ((beta + alpha) / i))));
	} else {
		tmp = (i / (fma(i, 2.0, (beta + alpha)) + 1.0)) * ((alpha + i) / ((beta + alpha) + fma(i, 2.0, -1.0)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) * 2.0)
	t_1 = Float64(beta + alpha) ^ 2.0
	tmp = 0.0
	if (beta <= 4.907639238008554e+137)
		tmp = Float64(Float64(Float64(fma(-2.0, Float64(Float64(beta + alpha) * fma(0.0625, t_0, Float64(Float64(beta + alpha) * -0.125))), Float64(fma(4.0, Float64(t_1 + -1.0), Float64(t_1 * 20.0)) * -0.00390625)) + Float64(0.0625 * Float64(t_1 + Float64(beta * alpha)))) / Float64(i * i)) + Float64(0.0625 + fma(0.0625, Float64(t_0 / i), Float64(-0.125 * Float64(Float64(beta + alpha) / i)))));
	else
		tmp = Float64(Float64(i / Float64(fma(i, 2.0, Float64(beta + alpha)) + 1.0)) * Float64(Float64(alpha + i) / Float64(Float64(beta + alpha) + fma(i, 2.0, -1.0))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 4.907639238008554e+137], N[(N[(N[(N[(-2.0 * N[(N[(beta + alpha), $MachinePrecision] * N[(0.0625 * t$95$0 + N[(N[(beta + alpha), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(t$95$1 + -1.0), $MachinePrecision] + N[(t$95$1 * 20.0), $MachinePrecision]), $MachinePrecision] * -0.00390625), $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[(t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(0.0625 * N[(t$95$0 / i), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9076392380085544e137

   1. Initial program 49.6

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   2. Taylor expanded in i around -inf 5.0

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{0.0625 \cdot \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) - \left(-2 \cdot \left(\left(0.0625 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.125 \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) + -0.00390625 \cdot \left(16 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + 4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right)\right)\right)\right)}{{i}^{2}} + \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   3. Simplified 5.0

      \[\leadsto \color{blue}{\frac{-\left(0.0625 \cdot \left(\beta \cdot \left(-\alpha\right) - {\left(\beta + \alpha\right)}^{2}\right) - \mathsf{fma}\left(-2, \left(\beta + \alpha\right) \cdot \mathsf{fma}\left(0.0625, 2 \cdot \left(\beta + \alpha\right), \left(\beta + \alpha\right) \cdot -0.125\right), \mathsf{fma}\left(4, {\left(\beta + \alpha\right)}^{2} + -1, {\left(\beta + \alpha\right)}^{2} \cdot 20\right) \cdot -0.00390625\right)\right)}{i \cdot i} + \left(0.0625 + \mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, \frac{\beta + \alpha}{i} \cdot -0.125\right)\right)} \]
      Proof
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (-.f64 (*.f64 beta (neg.f64 alpha)) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (*.f64 2 (+.f64 beta alpha)) (*.f64 (+.f64 beta alpha) -1/8))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (-.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 beta alpha))) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (*.f64 2 (+.f64 beta alpha)) (*.f64 (+.f64 beta alpha) -1/8))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 beta alpha))) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (*.f64 2 (+.f64 beta alpha)) (*.f64 (+.f64 beta alpha) -1/8))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (neg.f64 (pow.f64 (+.f64 beta alpha) 2))))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (*.f64 2 (+.f64 beta alpha)) (*.f64 (+.f64 beta alpha) -1/8))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2))))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (*.f64 2 (+.f64 beta alpha)) (*.f64 (+.f64 beta alpha) -1/8))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 (+.f64 beta alpha) -1/8))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) (*.f64 (+.f64 beta alpha) (Rewrite<= metadata-eval (neg.f64 1/8))))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (+.f64 beta alpha) 1/8))))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/8 (+.f64 beta alpha)))))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (fma.f64 1/16 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (neg.f64 (*.f64 1/8 (+.f64 beta alpha))))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) (Rewrite<= metadata-eval (neg.f64 1))) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (fma.f64 4 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20)) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (fma.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) (Rewrite<= metadata-eval (+.f64 16 4)))) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (fma.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (pow.f64 (+.f64 beta alpha) 2))))) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (pow.f64 (+.f64 beta alpha) 2))))) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))) (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)))) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))))) -1/256)))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (Rewrite<= *-commutative_binary64 (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (Rewrite=> fma-def_binary64 (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) 1/16)) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) 1/16) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 3 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))))) (*.f64 i i)) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)))))))) (Rewrite<= unpow2_binary64 (pow.f64 i 2))) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2)))) (+.f64 1/16 (fma.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (fma.f64 1/16 (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) i) (*.f64 (/.f64 (+.f64 beta alpha) i) -1/8)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (fma.f64 1/16 (/.f64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) i) (*.f64 (/.f64 (+.f64 beta alpha) i) (Rewrite<= metadata-eval (neg.f64 1/8)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (fma.f64 1/16 (/.f64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) i) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (+.f64 beta alpha) i) 1/8)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (fma.f64 1/16 (/.f64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) i) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/8 (/.f64 (+.f64 beta alpha) i))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)) i)) (*.f64 1/8 (/.f64 (+.f64 beta alpha) i)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (-.f64 (*.f64 1/16 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) i)) (*.f64 1/8 (/.f64 (+.f64 beta alpha) i))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1/16 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i))) (*.f64 1/8 (/.f64 (+.f64 beta alpha) i))))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))) (pow.f64 i 2))) (+.f64 1/16 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)))) (*.f64 1/8 (/.f64 (+.f64 beta alpha) i)))): 3 points increase in error, 0 points decrease in error

   if 4.9076392380085544e137 < beta

   1. Initial program 63.9

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   2. Taylor expanded in beta around inf 48.0

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

   3. Applied egg-rr 16.4

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i + \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \left(\beta + \alpha\right) \cdot \mathsf{fma}\left(0.0625, \left(\beta + \alpha\right) \cdot 2, \left(\beta + \alpha\right) \cdot -0.125\right), \mathsf{fma}\left(4, {\left(\beta + \alpha\right)}^{2} + -1, {\left(\beta + \alpha\right)}^{2} \cdot 20\right) \cdot -0.00390625\right) + 0.0625 \cdot \left({\left(\beta + \alpha\right)}^{2} + \beta \cdot \alpha\right)}{i \cdot i} + \left(0.0625 + \mathsf{fma}\left(0.0625, \frac{\left(\beta + \alpha\right) \cdot 2}{i}, -0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\alpha + i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.8
Cost	14276

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\alpha + i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternative 2
Error	16.0
Cost	1108

\[\begin{array}{l} t_0 := \frac{\frac{i \cdot i}{\beta}}{\beta}\\ \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.054620468810442 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 6.195782587374654 \cdot 10^{+203}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1986008444577911 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 3.06202612256522 \cdot 10^{+249}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha \cdot i}{\beta}}{\beta}\\ \end{array} \]

Alternative 3
Error	11.3
Cost	844

\[\begin{array}{l} t_0 := \frac{\frac{i}{\frac{\beta}{i}}}{\beta}\\ \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7.271168518329248 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 6.195782587374654 \cdot 10^{+203}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	11.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7.271168518329248 \cdot 10^{+189}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 6.195782587374654 \cdot 10^{+203}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i}}}{\beta}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\frac{\beta}{i}}}{\beta}\\ \end{array} \]

Alternative 6
Error	9.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + i\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 7
Error	9.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 8
Error	16.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.06202612256522 \cdot 10^{+249}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha \cdot i}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	10.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.907639238008554 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Error	18.1
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022290 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))