Octave 3.8, jcobi/4

Average Error: 54.2 → 2.1

Time: 24.7s

Precision: binary64

Cost: 8768

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \left(\frac{i}{t_0} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{t_0}}} \end{array} \]

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 (* i 2.0))))
   (*
    (* (/ i t_0) (/ (+ i beta) (+ (* i 2.0) (+ beta 1.0))))
    (/ i (/ (+ beta (fma i 2.0 -1.0)) (/ (+ i beta) t_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 + (i * 2.0);
	return ((i / t_0) * ((i + beta) / ((i * 2.0) + (beta + 1.0)))) * (i / ((beta + fma(i, 2.0, -1.0)) / ((i + beta) / t_0)));
}

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(beta + Float64(i * 2.0))
	return Float64(Float64(Float64(i / t_0) * Float64(Float64(i + beta) / Float64(Float64(i * 2.0) + Float64(beta + 1.0)))) * Float64(i / Float64(Float64(beta + fma(i, 2.0, -1.0)) / Float64(Float64(i + beta) / t_0))))
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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(N[(i * 2.0), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(beta + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(i + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.2

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

2. Applied egg-rr 37.5

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

3. Taylor expanded in alpha around 0 39.7

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

4. Simplified 2.1

   \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \color{blue}{\frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{\beta + i}{\beta + i \cdot 2}}}} \]
   Proof
   (/.f64 i (/.f64 (+.f64 beta (fma.f64 i 2 -1)) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 beta (fma.f64 i 2 (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 beta (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 i 2) 1))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 beta (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) 1)) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 beta (*.f64 2 i)) 1)) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (/.f64 (+.f64 beta i) (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (+.f64 beta (*.f64 2 i))) (+.f64 beta i)))): 144 points increase in error, 24 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (+.f64 beta i)) (*.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (+.f64 beta (*.f64 2 i))))): 153 points increase in error, 7 points decrease in error

5. Taylor expanded in alpha around 0 39.7

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

6. Simplified 2.1

   \[\leadsto \color{blue}{\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\left(\beta + 1\right) + i \cdot 2}\right)} \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{\beta + i}{\beta + i \cdot 2}}} \]
   Proof
   (*.f64 (/.f64 i (+.f64 beta (*.f64 i 2))) (/.f64 (+.f64 beta i) (+.f64 (+.f64 beta 1) (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 i (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i)))) (/.f64 (+.f64 beta i) (+.f64 (+.f64 beta 1) (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 i (+.f64 beta (*.f64 2 i))) (/.f64 (+.f64 beta i) (+.f64 (+.f64 beta 1) (Rewrite<= *-commutative_binary64 (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 i (+.f64 beta (*.f64 2 i))) (/.f64 (+.f64 beta i) (Rewrite<= associate-+r+_binary64 (+.f64 beta (+.f64 1 (*.f64 2 i)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 beta i)) (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 1 (*.f64 2 i)))))): 157 points increase in error, 22 points decrease in error

7. Final simplification 2.1

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

Alternatives

Alternative 1
Error	9.9
Cost	8776

\[\begin{array}{l} t_0 := \frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\\ \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;t_0 \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{1 - \frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;t_0 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 2
Error	9.9
Cost	8776

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \frac{i}{t_0}\\ t_2 := t_1 \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\\ t_3 := \beta + \mathsf{fma}\left(i, 2, -1\right)\\ \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;\frac{i}{\frac{t_3}{\frac{i + \beta}{t_0}}} \cdot \left(t_1 \cdot \left(0.5 + \frac{\beta \cdot 0.25 + -0.25}{i}\right)\right)\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;t_2 \cdot \frac{i}{\frac{t_3}{1 - \frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;t_2 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	8264

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \frac{i}{t_0}\\ \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;t_1 \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{t_0}}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;\left(t_1 \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Error	9.8
Cost	8008

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{t_0}}} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	1740

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	9.8
Cost	1740

\[\begin{array}{l} t_0 := \frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\\ \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;t_0 \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;t_0 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 7
Error	10.0
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{i}{\frac{\beta}{\frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i + \alpha}}}{\beta}\\ \end{array} \]

Alternative 8
Error	10.0
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i + \alpha}}}{\beta}\\ \end{array} \]

Alternative 9
Error	10.0
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	9.9
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 11
Error	16.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 9.297695384145658 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Error	15.7
Cost	844

\[\begin{array}{l} t_0 := \frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	11.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	11.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{i}{\frac{\beta}{\frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Error	11.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6556421478494 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.477508983875876 \cdot 10^{+136}:\\ \;\;\;\;\frac{i}{\frac{\beta}{\frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 8.500138643441153 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 16
Error	16.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.297695384145658 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	19.3
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022294 
(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)))