Average Error: 54.3 → 12.2
Time: 1.0min
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 2.7053834425805328 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{elif}\;i \leq 1.80215627512591 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 8.438851657495755 \cdot 10^{+119}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + \left(i + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
\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 2.7053834425805328 \cdot 10^{+59}:\\
\;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}\\

\mathbf{elif}\;i \leq 1.80215627512591 \cdot 10^{+78}:\\
\;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;i \leq 8.438851657495755 \cdot 10^{+119}:\\
\;\;\;\;\frac{i \cdot \frac{\alpha + \left(i + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \cdot \frac{\frac{\alpha \cdot \beta + 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}\\

\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 2.7053834425805328e+59)
   (/
    (/ (* i (+ i (+ alpha beta))) (+ (+ alpha beta) (* i 2.0)))
    (/
     (- (* (+ (+ alpha beta) (* i 2.0)) (+ (+ alpha beta) (* i 2.0))) 1.0)
     (/
      (+ (* i (+ i (+ alpha beta))) (* alpha beta))
      (+ (+ alpha beta) (* i 2.0)))))
   (if (<= i 1.80215627512591e+78)
     (/
      (* (* i i) 0.25)
      (- (* (+ (+ alpha beta) (* i 2.0)) (+ (+ alpha beta) (* i 2.0))) 1.0))
     (if (<= i 8.438851657495755e+119)
       (*
        (/
         (* i (/ (+ alpha (+ i beta)) (+ alpha (+ beta (* i 2.0)))))
         (+ 1.0 (+ alpha (+ beta (* i 2.0)))))
        (/
         (/
          (+ (* alpha beta) (* i (+ alpha (+ i 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 <= 2.7053834425805328e+59) {
		tmp = ((i * (i + (alpha + beta))) / ((alpha + beta) + (i * 2.0))) / (((((alpha + beta) + (i * 2.0)) * ((alpha + beta) + (i * 2.0))) - 1.0) / (((i * (i + (alpha + beta))) + (alpha * beta)) / ((alpha + beta) + (i * 2.0))));
	} else if (i <= 1.80215627512591e+78) {
		tmp = ((i * i) * 0.25) / ((((alpha + beta) + (i * 2.0)) * ((alpha + beta) + (i * 2.0))) - 1.0);
	} else if (i <= 8.438851657495755e+119) {
		tmp = ((i * ((alpha + (i + beta)) / (alpha + (beta + (i * 2.0))))) / (1.0 + (alpha + (beta + (i * 2.0))))) * ((((alpha * beta) + (i * (alpha + (i + 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 4 regimes

  2. if i < 2.7053834425805328e59

    1. Initial program 23.7

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

    3. Applied times-frac_binary64_178911.4

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

    4. Applied associate-/l*_binary64_172811.4

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

    if 2.7053834425805328e59 < i < 1.80215627512591e78

    1. Initial program 30.9

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

    2. Taylor expanded around inf 18.1

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

    3. Simplified18.1

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

    if 1.80215627512591e78 < i < 8.4388516574957552e119

    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. Using strategy rm

    3. Applied difference-of-sqr-1_binary64_175364.0

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

    4. Applied times-frac_binary64_178920.4

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

    5. Applied times-frac_binary64_178914.0

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

    6. Simplified14.0

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

    7. Simplified14.0

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

    9. Applied *-un-lft-identity_binary64_178314.0

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

    10. Applied times-frac_binary64_178913.9

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

    if 8.4388516574957552e119 < 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 11.5

      \[\leadsto \color{blue}{0.0625}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.7053834425805328 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{elif}\;i \leq 1.80215627512591 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 8.438851657495755 \cdot 10^{+119}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + \left(i + \beta\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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)))