Average Error: 54.0 → 10.8
Time: 32.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 1.4261104219147907 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)} \cdot \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + \left(0.0625 - 0.03125 \cdot \left(\beta \cdot \frac{\beta}{i \cdot i}\right)\right)\\ \end{array}\]
\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 1.4261104219147907 \cdot 10^{+153}:\\
\;\;\;\;\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)} \cdot \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.015625}{i \cdot i} + \left(0.0625 - 0.03125 \cdot \left(\beta \cdot \frac{\beta}{i \cdot i}\right)\right)\\

\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 1.4261104219147907e+153)
   (*
    (/
     (sqrt (+ (* i (+ beta (+ i alpha))) (* beta alpha)))
     (+ (+ alpha (+ beta (* i 2.0))) 1.0))
    (*
     (/
      (sqrt (+ (* i (+ beta (+ i alpha))) (* beta alpha)))
      (+ alpha (+ (+ beta (* i 2.0)) -1.0)))
     (*
      (/ i (+ alpha (+ beta (* i 2.0))))
      (/ (+ beta (+ i alpha)) (+ alpha (+ beta (* i 2.0)))))))
   (+ (/ 0.015625 (* i i)) (- 0.0625 (* 0.03125 (* beta (/ beta (* i i))))))))
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 <= 1.4261104219147907e+153) {
		tmp = (sqrt((i * (beta + (i + alpha))) + (beta * alpha)) / ((alpha + (beta + (i * 2.0))) + 1.0)) * ((sqrt((i * (beta + (i + alpha))) + (beta * alpha)) / (alpha + ((beta + (i * 2.0)) + -1.0))) * ((i / (alpha + (beta + (i * 2.0)))) * ((beta + (i + alpha)) / (alpha + (beta + (i * 2.0))))));
	} else {
		tmp = (0.015625 / (i * i)) + (0.0625 - (0.03125 * (beta * (beta / (i * i)))));
	}
	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 < 1.4261104219147907e153

    1. Initial program 43.6

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

    2. Simplified16.2

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

    4. Applied difference-of-sqr-1_binary64_243516.2

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

    5. Applied add-sqr-sqrt_binary64_248716.2

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

    6. Applied times-frac_binary64_247116.2

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

    7. Applied associate-*l*_binary64_240616.2

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

    8. Simplified11.7

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

    if 1.4261104219147907e153 < 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. Simplified63.8

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

    3. Taylor expanded around inf 19.7

      \[\leadsto \color{blue}{\left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right) - \left(0.03125 \cdot \frac{{\alpha}^{2}}{{i}^{2}} + 0.03125 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)}\]

    4. Simplified19.7

      \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + \left(0.0625 - 0.03125 \cdot \left(\frac{\beta \cdot \beta}{i \cdot i} + \frac{\alpha \cdot \alpha}{i \cdot i}\right)\right)}\]

    5. Taylor expanded around inf 15.6

      \[\leadsto \frac{0.015625}{i \cdot i} + \left(0.0625 - 0.03125 \cdot \color{blue}{\frac{{\beta}^{2}}{{i}^{2}}}\right)\]

    6. Simplified9.9

      \[\leadsto \frac{0.015625}{i \cdot i} + \left(0.0625 - 0.03125 \cdot \color{blue}{\left(\beta \cdot \frac{\beta}{i \cdot i}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.4261104219147907 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)} \cdot \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \frac{\beta + \left(i + \alpha\right)}{\alpha + \left(\beta + i \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + \left(0.0625 - 0.03125 \cdot \left(\beta \cdot \frac{\beta}{i \cdot i}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))