Average Error: 54.0 → 14.7
Time: 6.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.5824841192270528 \cdot 10^{+111} \lor \neg \left(\beta \leq 1.202295370415579 \cdot 10^{+144}\right) \land \beta \leq 2.178716223398337 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{i}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{i + \left(\beta + \alpha\right)}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{i + \left(\beta + \alpha\right)}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\ \end{array}\]
\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}\;\beta \leq 1.5824841192270528 \cdot 10^{+111} \lor \neg \left(\beta \leq 1.202295370415579 \cdot 10^{+144}\right) \land \beta \leq 2.178716223398337 \cdot 10^{+180}:\\
\;\;\;\;\frac{\frac{i}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{i + \left(\beta + \alpha\right)}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{i}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{i + \left(\beta + \alpha\right)}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\

\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 (or (<= beta 1.5824841192270528e+111)
         (and (not (<= beta 1.202295370415579e+144))
              (<= beta 2.178716223398337e+180)))
   (*
    (/
     (*
      (/ i (sqrt (+ (+ beta alpha) (* i 2.0))))
      (/ (+ i (+ beta alpha)) (sqrt (+ (+ beta alpha) (* i 2.0)))))
     (+ (+ (+ beta alpha) (* i 2.0)) (sqrt 1.0)))
    (/
     (+ (* i 0.5) (* (+ beta alpha) 0.25))
     (- (+ (+ beta alpha) (* i 2.0)) (sqrt 1.0))))
   (*
    (/
     (*
      (/ i (sqrt (+ (+ beta alpha) (* i 2.0))))
      (/ (+ i (+ beta alpha)) (sqrt (+ (+ beta alpha) (* i 2.0)))))
     (+ (+ (+ beta alpha) (* i 2.0)) (sqrt 1.0)))
    (/ i (- (+ (+ beta alpha) (* i 2.0)) (sqrt 1.0))))))
double code(double alpha, double beta, double i) {
	return ((((double) (((double) (i * ((double) (((double) (alpha + beta)) + i)))) * ((double) (((double) (beta * alpha)) + ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i))))))) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) - 1.0)));
}
↓
double code(double alpha, double beta, double i) {
	double tmp;
	if (((beta <= 1.5824841192270528e+111) || (!(beta <= 1.202295370415579e+144) && (beta <= 2.178716223398337e+180)))) {
		tmp = ((double) ((((double) ((i / ((double) sqrt(((double) (((double) (beta + alpha)) + ((double) (i * 2.0))))))) * (((double) (i + ((double) (beta + alpha)))) / ((double) sqrt(((double) (((double) (beta + alpha)) + ((double) (i * 2.0))))))))) / ((double) (((double) (((double) (beta + alpha)) + ((double) (i * 2.0)))) + ((double) sqrt(1.0))))) * (((double) (((double) (i * 0.5)) + ((double) (((double) (beta + alpha)) * 0.25)))) / ((double) (((double) (((double) (beta + alpha)) + ((double) (i * 2.0)))) - ((double) sqrt(1.0)))))));
	} else {
		tmp = ((double) ((((double) ((i / ((double) sqrt(((double) (((double) (beta + alpha)) + ((double) (i * 2.0))))))) * (((double) (i + ((double) (beta + alpha)))) / ((double) sqrt(((double) (((double) (beta + alpha)) + ((double) (i * 2.0))))))))) / ((double) (((double) (((double) (beta + alpha)) + ((double) (i * 2.0)))) + ((double) sqrt(1.0))))) * (i / ((double) (((double) (((double) (beta + alpha)) + ((double) (i * 2.0)))) - ((double) sqrt(1.0)))))));
	}
	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 beta < 1.5824841192270528e111 or 1.20229537041557906e144 < beta < 2.17871622339833693e180

    1. Initial program 52.1

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

    3. Applied add-sqr-sqrt_binary6452.1

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

    4. Applied difference-of-squares_binary6452.1

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

    5. Applied times-frac_binary6436.7

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

    6. Applied times-frac_binary6434.7

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

    7. Simplified34.7

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

    8. Simplified34.7

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

    10. Applied add-sqr-sqrt_binary6434.8

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

    11. Applied times-frac_binary6434.8

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

    12. Taylor expanded around 0 13.0

      \[\leadsto \frac{\frac{i}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}} \cdot \frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\]

    13. Simplified13.0

      \[\leadsto \frac{\frac{i}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}} \cdot \frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{\color{blue}{i \cdot 0.5 + 0.25 \cdot \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\]

    if 1.5824841192270528e111 < beta < 1.20229537041557906e144 or 2.17871622339833693e180 < beta

    1. Initial program 62.8

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

    3. Applied add-sqr-sqrt_binary6462.8

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

    4. Applied difference-of-squares_binary6462.8

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

    5. Applied times-frac_binary6452.8

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

    6. Applied times-frac_binary6450.1

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

    7. Simplified50.1

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

    8. Simplified50.1

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

    10. Applied add-sqr-sqrt_binary6450.1

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

    11. Applied times-frac_binary6450.1

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

    12. Taylor expanded around inf 22.5

      \[\leadsto \frac{\frac{i}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}} \cdot \frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{\color{blue}{i}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5824841192270528 \cdot 10^{+111} \lor \neg \left(\beta \leq 1.202295370415579 \cdot 10^{+144}\right) \land \beta \leq 2.178716223398337 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{i}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{i + \left(\beta + \alpha\right)}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \frac{i + \left(\beta + \alpha\right)}{\sqrt{\left(\beta + \alpha\right) + i \cdot 2}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))