Average Error: 53.7 → 10.8
Time: 26.1s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 8.623959790867124 \cdot 10^{+146}:\\ \;\;\;\;\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + \sqrt{1}} \cdot \frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha} \cdot \left(\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta + \left(i + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
\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 8.623959790867124 \cdot 10^{+146}:\\
\;\;\;\;\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + \sqrt{1}} \cdot \frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha} \cdot \left(\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta + \left(i + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) - \sqrt{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 8.623959790867124e+146)
   (*
    (/
     (sqrt (+ (* i (+ beta (+ i alpha))) (* beta alpha)))
     (+ (+ alpha (+ beta (* i 2.0))) (sqrt 1.0)))
    (/
     (*
      (sqrt (+ (* i (+ beta (+ i alpha))) (* beta alpha)))
      (*
       (/ i (+ (* i 2.0) (+ beta alpha)))
       (/ (+ beta (+ i alpha)) (+ (* i 2.0) (+ beta alpha)))))
     (- (+ (* i 2.0) (+ beta alpha)) (sqrt 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 <= 8.623959790867124e+146) {
		tmp = (sqrt((i * (beta + (i + alpha))) + (beta * alpha)) / ((alpha + (beta + (i * 2.0))) + sqrt(1.0))) * ((sqrt((i * (beta + (i + alpha))) + (beta * alpha)) * ((i / ((i * 2.0) + (beta + alpha))) * ((beta + (i + alpha)) / ((i * 2.0) + (beta + alpha))))) / (((i * 2.0) + (beta + alpha)) - sqrt(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 < 8.6239597908671239e146

    1. Initial program 43.0

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

    2. Simplified15.3

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

    4. Applied add-sqr-sqrt_binary64_180515.3

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

    5. Applied difference-of-squares_binary64_175215.3

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

    6. Applied add-sqr-sqrt_binary64_180515.3

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

    7. Applied times-frac_binary64_178915.3

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

    8. Applied associate-*l*_binary64_172415.3

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

    9. Simplified11.1

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

    if 8.6239597908671239e146 < 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. Simplified62.1

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

    3. Taylor expanded around inf 10.4

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 8.623959790867124 \cdot 10^{+146}:\\ \;\;\;\;\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + \sqrt{1}} \cdot \frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha} \cdot \left(\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta + \left(i + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2021210 
(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)))