Average Error: 54.1 → 11.4
Time: 5.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 4.260945402677668 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i \cdot e^{\log \left(\frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}}\right)}\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\\ \end{array}\]
\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}\;i \leq 4.260945402677668 \cdot 10^{+49}:\\
\;\;\;\;\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{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(i \cdot e^{\log \left(\frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}}\right)}\right) \cdot 0.25}{\left(\left(\alpha + \beta\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 (<= i 4.260945402677668e+49)
   (*
    (/
     (/ (* i (+ i (+ alpha beta))) (+ (+ alpha beta) (* i 2.0)))
     (+ (+ (+ alpha beta) (* i 2.0)) (sqrt 1.0)))
    (/
     (/
      (+ (* i (+ i (+ alpha beta))) (* alpha beta))
      (+ (+ alpha beta) (* i 2.0)))
     (- (+ (+ alpha beta) (* i 2.0)) (sqrt 1.0))))
   (/
    (*
     (* i (exp (log (/ i (+ (+ (+ alpha beta) (* i 2.0)) (sqrt 1.0))))))
     0.25)
    (- (+ (+ alpha beta) (* 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 ((i <= 4.260945402677668e+49)) {
		tmp = ((double) (((((double) (i * ((double) (i + ((double) (alpha + beta)))))) / ((double) (((double) (alpha + beta)) + ((double) (i * 2.0))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (i * 2.0)))) + ((double) sqrt(1.0))))) * ((((double) (((double) (i * ((double) (i + ((double) (alpha + beta)))))) + ((double) (alpha * beta)))) / ((double) (((double) (alpha + beta)) + ((double) (i * 2.0))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (i * 2.0)))) - ((double) sqrt(1.0)))))));
	} else {
		tmp = (((double) (((double) (i * ((double) exp(((double) log((i / ((double) (((double) (((double) (alpha + beta)) + ((double) (i * 2.0)))) + ((double) sqrt(1.0))))))))))) * 0.25)) / ((double) (((double) (((double) (alpha + beta)) + ((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 i < 4.26094540267766779e49

    1. Initial program 22.1

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

    3. Applied add-sqr-sqrt_binary6422.1

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

    4. Applied difference-of-squares_binary6422.1

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

    5. Applied times-frac_binary649.3

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

    6. Applied times-frac_binary646.5

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

    7. Simplified6.5

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

    8. Simplified6.5

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

    if 4.26094540267766779e49 < i

    1. Initial program 60.1

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

    2. Taylor expanded around inf 45.2

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

    3. Simplified45.2

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

    5. Applied add-sqr-sqrt_binary6445.2

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

    6. Applied difference-of-squares_binary6445.2

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

    7. Applied associate-/r*_binary6444.7

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

    8. Simplified44.7

      \[\leadsto \frac{\color{blue}{\frac{i \cdot i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}} \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity_binary6444.7

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

    11. Applied times-frac_binary6412.3

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

    12. Simplified12.3

      \[\leadsto \frac{\left(\color{blue}{i} \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}}\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
    13. Using strategy rm

    14. Applied add-exp-log_binary6416.9

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

    15. Applied add-exp-log_binary6416.7

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

    16. Applied div-exp_binary6416.7

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

    17. Simplified12.3

      \[\leadsto \frac{\left(i \cdot e^{\color{blue}{\log \left(\frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}}\right)}}\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.260945402677668 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i \cdot e^{\log \left(\frac{i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + \sqrt{1}}\right)}\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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)))