Average Error: 54.3 → 11.4
Time: 45.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[[alpha, beta]=\mathsf{sort}([alpha, beta])\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.152445749605528 \cdot 10^{+97}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7.353764786069445 \cdot 10^{+139}:\\ \;\;\;\;i \cdot \left(\frac{\frac{\beta \cdot \alpha + i \cdot \left(\alpha + \left(\beta + i\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \cdot \frac{\frac{\alpha + \left(\beta + i\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)}\right)\\ \mathbf{elif}\;\beta \leq 3.2397109221427535 \cdot 10^{+212}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i + \alpha}}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)}\\ \end{array}\]
\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}\;\beta \leq 6.152445749605528 \cdot 10^{+97}:\\
\;\;\;\;0.0625\\

\mathbf{elif}\;\beta \leq 7.353764786069445 \cdot 10^{+139}:\\
\;\;\;\;i \cdot \left(\frac{\frac{\beta \cdot \alpha + i \cdot \left(\alpha + \left(\beta + i\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \cdot \frac{\frac{\alpha + \left(\beta + i\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)}\right)\\

\mathbf{elif}\;\beta \leq 3.2397109221427535 \cdot 10^{+212}:\\
\;\;\;\;0.0625\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{i}{\frac{\beta}{i + \alpha}}}{1 + \left(\alpha + \left(\beta + i \cdot 2\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 (<= beta 6.152445749605528e+97)
   0.0625
   (if (<= beta 7.353764786069445e+139)
     (*
      i
      (*
       (/
        (/
         (+ (* beta alpha) (* i (+ alpha (+ beta i))))
         (+ alpha (+ beta (* i 2.0))))
        (+ alpha (- (+ beta (* i 2.0)) 1.0)))
       (/
        (/ (+ alpha (+ beta i)) (+ alpha (+ beta (* i 2.0))))
        (+ 1.0 (+ alpha (+ beta (* i 2.0)))))))
     (if (<= beta 3.2397109221427535e+212)
       0.0625
       (/ (/ i (/ beta (+ i alpha))) (+ 1.0 (+ alpha (+ beta (* i 2.0)))))))))
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 (beta <= 6.152445749605528e+97) {
		tmp = 0.0625;
	} else if (beta <= 7.353764786069445e+139) {
		tmp = i * (((((beta * alpha) + (i * (alpha + (beta + i)))) / (alpha + (beta + (i * 2.0)))) / (alpha + ((beta + (i * 2.0)) - 1.0))) * (((alpha + (beta + i)) / (alpha + (beta + (i * 2.0)))) / (1.0 + (alpha + (beta + (i * 2.0))))));
	} else if (beta <= 3.2397109221427535e+212) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta / (i + alpha))) / (1.0 + (alpha + (beta + (i * 2.0))));
	}
	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 3 regimes

  2. if beta < 6.152445749605528e97 or 7.3537647860694452e139 < beta < 3.2397109221427535e212

    1. Initial program 51.2

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

    2. Taylor expanded around inf 9.3

      \[\leadsto \color{blue}{0.0625}\]

    if 6.152445749605528e97 < beta < 7.3537647860694452e139

    1. Initial program 58.6

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

    3. Applied difference-of-sqr-1_binary64_209458.6

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

    4. Applied times-frac_binary64_213032.0

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

    5. Applied times-frac_binary64_213031.9

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

    6. Simplified31.9

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

    7. Simplified32.0

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

    9. Applied *-un-lft-identity_binary64_212432.0

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

    10. Applied *-un-lft-identity_binary64_212432.0

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

    11. Applied times-frac_binary64_213031.9

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

    12. Applied times-frac_binary64_213031.9

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

    13. Applied associate-*l*_binary64_206531.9

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

    14. Simplified31.9

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

    if 3.2397109221427535e212 < beta

    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_209464.0

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

    4. Applied times-frac_binary64_213056.7

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

    5. Applied times-frac_binary64_213056.3

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

    6. Simplified56.3

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

    7. Simplified56.3

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

    9. Applied associate-*l/_binary64_206756.3

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

    10. Simplified56.3

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

    11. Taylor expanded around inf 32.4

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

    12. Simplified11.1

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.152445749605528 \cdot 10^{+97}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7.353764786069445 \cdot 10^{+139}:\\ \;\;\;\;i \cdot \left(\frac{\frac{\beta \cdot \alpha + i \cdot \left(\alpha + \left(\beta + i\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \cdot \frac{\frac{\alpha + \left(\beta + i\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)}\right)\\ \mathbf{elif}\;\beta \leq 3.2397109221427535 \cdot 10^{+212}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i + \alpha}}}{1 + \left(\alpha + \left(\beta + i \cdot 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(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)))