Average Error: 54.2 → 11.3
Time: 20.6s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 3.262397731415471 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
\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}
\mathbf{if}\;i \leq 3.262397731415471 \cdot 10^{+126}:\\
\;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;0.0625\\

\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
 (if (<= i 3.262397731415471e+126)
   (*
    (/
     (/ (* i (+ alpha (+ i beta))) (+ alpha (+ beta (* i 2.0))))
     (+ (+ alpha (+ beta (* i 2.0))) 1.0))
    (/
     (/
      (+ (* i (+ alpha (+ i beta))) (* alpha beta))
      (+ alpha (+ beta (* i 2.0))))
     (- (+ alpha (+ beta (* i 2.0))) 1.0)))
   0.0625))
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 tmp;
	if (i <= 3.262397731415471e+126) {
		tmp = (((i * (alpha + (i + beta))) / (alpha + (beta + (i * 2.0)))) / ((alpha + (beta + (i * 2.0))) + 1.0)) * ((((i * (alpha + (i + beta))) + (alpha * beta)) / (alpha + (beta + (i * 2.0)))) / ((alpha + (beta + (i * 2.0))) - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 3.2623977314154708e126

    1. Initial program 39.9

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

    3. Applied difference-of-sqr-1_binary64_141239.9

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

    4. Applied times-frac_binary64_144814.8

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

    5. Applied times-frac_binary64_144810.2

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

    6. Simplified10.2

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

    7. Simplified10.2

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

    if 3.2623977314154708e126 < i

    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 around inf 12.0

      \[\leadsto \color{blue}{0.0625}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.262397731415471 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (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)))