Average Error: 53.9 → 8.9
Time: 21.9s
Precision: binary64
Cost: 28364
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := t_0 + -1\\ t_2 := \frac{\frac{\beta}{\frac{t_0}{\beta}}}{1 + t_0} \cdot \frac{\frac{i}{\frac{t_0}{i + \alpha}}}{t_1}\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{0.25 \cdot i}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 1\right)\right)}{i}}}{t_1}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+215}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{0.25 \cdot i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
(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 (fma i 2.0 (+ beta alpha)))
        (t_1 (+ t_0 -1.0))
        (t_2
         (*
          (/ (/ beta (/ t_0 beta)) (+ 1.0 t_0))
          (/ (/ i (/ t_0 (+ i alpha))) t_1))))
   (if (<= beta 1.8e+163)
     (/ (/ (* 0.25 i) (/ (+ beta (+ alpha (fma i 2.0 1.0))) i)) t_1)
     (if (<= beta 6.2e+198)
       t_2
       (if (<= beta 6.2e+215)
         (*
          (/ i (+ (fma i 2.0 1.0) (+ beta alpha)))
          (/ (* 0.25 i) (+ (+ beta alpha) (fma i 2.0 -1.0))))
         t_2)))))
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 = fma(i, 2.0, (beta + alpha));
	double t_1 = t_0 + -1.0;
	double t_2 = ((beta / (t_0 / beta)) / (1.0 + t_0)) * ((i / (t_0 / (i + alpha))) / t_1);
	double tmp;
	if (beta <= 1.8e+163) {
		tmp = ((0.25 * i) / ((beta + (alpha + fma(i, 2.0, 1.0))) / i)) / t_1;
	} else if (beta <= 6.2e+198) {
		tmp = t_2;
	} else if (beta <= 6.2e+215) {
		tmp = (i / (fma(i, 2.0, 1.0) + (beta + alpha))) * ((0.25 * i) / ((beta + alpha) + fma(i, 2.0, -1.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(Float64(Float64(beta / Float64(t_0 / beta)) / Float64(1.0 + t_0)) * Float64(Float64(i / Float64(t_0 / Float64(i + alpha))) / t_1))
	tmp = 0.0
	if (beta <= 1.8e+163)
		tmp = Float64(Float64(Float64(0.25 * i) / Float64(Float64(beta + Float64(alpha + fma(i, 2.0, 1.0))) / i)) / t_1);
	elseif (beta <= 6.2e+198)
		tmp = t_2;
	elseif (beta <= 6.2e+215)
		tmp = Float64(Float64(i / Float64(fma(i, 2.0, 1.0) + Float64(beta + alpha))) * Float64(Float64(0.25 * i) / Float64(Float64(beta + alpha) + fma(i, 2.0, -1.0))));
	else
		tmp = t_2;
	end
	return tmp
end

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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(beta / N[(t$95$0 / beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(t$95$0 / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.8e+163], N[(N[(N[(0.25 * i), $MachinePrecision] / N[(N[(beta + N[(alpha + N[(i * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[beta, 6.2e+198], t$95$2, If[LessEqual[beta, 6.2e+215], N[(N[(i / N[(N[(i * 2.0 + 1.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * i), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

\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 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\
t_1 := t_0 + -1\\
t_2 := \frac{\frac{\beta}{\frac{t_0}{\beta}}}{1 + t_0} \cdot \frac{\frac{i}{\frac{t_0}{i + \alpha}}}{t_1}\\
\mathbf{if}\;\beta \leq 1.8 \cdot 10^{+163}:\\
\;\;\;\;\frac{\frac{0.25 \cdot i}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 1\right)\right)}{i}}}{t_1}\\

\mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+198}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+215}:\\
\;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{0.25 \cdot i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if beta < 1.79999999999999989e163

    1. Initial program 50.0

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

    2. Taylor expanded in i around inf 38.0

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    3. Simplified38.0

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (*.f64 (*.f64 i i) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) 1/4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (pow.f64 i 2))): 0 points increase in error, 0 points decrease in error

    4. Applied egg-rr6.0

      \[\leadsto \color{blue}{\frac{i}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i \cdot 0.25}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}} \]

    5. Simplified6.0

      \[\leadsto \color{blue}{\frac{\frac{0.25 \cdot i}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 1\right)\right)}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}} \]
      Proof
      (/.f64 (/.f64 (*.f64 1/4 i) (/.f64 (+.f64 beta (+.f64 alpha (fma.f64 i 2 1))) i)) (+.f64 (fma.f64 i 2 (+.f64 beta alpha)) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 i 1/4)) (/.f64 (+.f64 beta (+.f64 alpha (fma.f64 i 2 1))) i)) (+.f64 (fma.f64 i 2 (+.f64 beta alpha)) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i 1/4) (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) (fma.f64 i 2 1))) i)) (+.f64 (fma.f64 i 2 (+.f64 beta alpha)) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i 1/4) (/.f64 (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta)) (fma.f64 i 2 1)) i)) (+.f64 (fma.f64 i 2 (+.f64 beta alpha)) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 i (/.f64 (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1)) i)) 1/4)) (+.f64 (fma.f64 i 2 (+.f64 beta alpha)) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1)))) 1/4) (+.f64 (fma.f64 i 2 (+.f64 beta alpha)) -1)): 150 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (fma.f64 i 2 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 i 2) alpha) beta)) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (*.f64 i 2))) beta) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (Rewrite=> associate-+l+_binary64 (+.f64 alpha (+.f64 (*.f64 i 2) beta))) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 i 2)))) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 i 2))) -1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (Rewrite<= associate-+r+_binary64 (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 i 2) -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) 1/4) (+.f64 (+.f64 alpha beta) (Rewrite<= fma-udef_binary64 (fma.f64 i 2 -1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 (/.f64 (*.f64 i i) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) (/.f64 (+.f64 (+.f64 alpha beta) (fma.f64 i 2 -1)) 1/4))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 i (/.f64 (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1)) i))) (/.f64 (+.f64 (+.f64 alpha beta) (fma.f64 i 2 -1)) 1/4)): 4 points increase in error, 150 points decrease in error
      (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) i)) (/.f64 (+.f64 (+.f64 alpha beta) (fma.f64 i 2 -1)) 1/4)): 7 points increase in error, 7 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) (/.f64 i (/.f64 (+.f64 (+.f64 alpha beta) (fma.f64 i 2 -1)) 1/4)))): 15 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (fma.f64 i 2 1))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i 1/4) (+.f64 (+.f64 alpha beta) (fma.f64 i 2 -1))))): 0 points increase in error, 1 points decrease in error

    if 1.79999999999999989e163 < beta < 6.1999999999999995e198 or 6.1999999999999998e215 < beta

    1. Initial program 64.0

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

    2. Taylor expanded in beta around inf 64.0

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

    3. Simplified64.0

      \[\leadsto \frac{\frac{\color{blue}{\left(\beta \cdot \beta\right) \cdot \left(i \cdot \left(i + \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (*.f64 (*.f64 beta beta) (*.f64 i (+.f64 i alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (*.f64 i (+.f64 i alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 beta 2) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 i alpha) i))): 0 points increase in error, 0 points decrease in error

    4. Applied egg-rr14.8

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

    if 6.1999999999999995e198 < beta < 6.1999999999999998e215

    1. Initial program 64.0

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

    2. Taylor expanded in i around inf 52.2

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    3. Simplified52.2

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (*.f64 (*.f64 i i) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) 1/4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (pow.f64 i 2))): 0 points increase in error, 0 points decrease in error

    4. Applied egg-rr28.5

      \[\leadsto \color{blue}{\frac{i}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i \cdot 0.25}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{0.25 \cdot i}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 1\right)\right)}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\beta}}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \alpha}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+215}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{0.25 \cdot i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\beta}}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \alpha}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]

Alternatives

Alternative 1
Error9.0
Cost14540
\[\begin{array}{l} t_0 := \frac{i}{\mathsf{fma}\left(i, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{0.25 \cdot i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ t_1 := \frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 5.6 \cdot 10^{+217}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error8.9
Cost14540
\[\begin{array}{l} t_0 := \frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{0.25 \cdot i}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 1\right)\right)}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+215}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{0.25 \cdot i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error16.1
Cost708
\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+230}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta} \cdot \frac{i \cdot i}{\beta}\\ \end{array} \]
Alternative 4
Error9.2
Cost708
\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+145}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
Alternative 5
Error16.0
Cost580
\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+232}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 6
Error16.2
Cost196
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+230}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 7
Error57.7
Cost64
\[0 \]

Error

Reproduce

herbie shell --seed 2022339 
(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)))