Average Error: 54.4 → 35.5
Time: 6.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.795987046005015 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \end{array}\]
\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 7.795987046005015 \cdot 10^{+157}:\\
\;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 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 (<= beta 7.795987046005015e+157)
   (/
    (*
     (/
      (+ (* beta alpha) (* i (+ i (+ beta alpha))))
      (+ (+ beta alpha) (* i 2.0)))
     (/
      (/ (* i (+ i (+ beta alpha))) (+ (+ beta alpha) (* i 2.0)))
      (+ (+ (+ beta alpha) (* i 2.0)) 1.0)))
    (- (+ (+ beta alpha) (* i 2.0)) 1.0))
   (/
    (*
     (* i i)
     (/
      (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0)))
      (+ (+ (+ beta alpha) (* i 2.0)) 1.0)))
    (- (+ (+ beta alpha) (* i 2.0)) 1.0))))
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 <= 7.795987046005015e+157) {
		tmp = ((((beta * alpha) + (i * (i + (beta + alpha)))) / ((beta + alpha) + (i * 2.0))) * (((i * (i + (beta + alpha))) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) + 1.0))) / (((beta + alpha) + (i * 2.0)) - 1.0);
	} else {
		tmp = ((i * i) * (((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) + 1.0))) / (((beta + alpha) + (i * 2.0)) - 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 < 7.7959870460050154e157

    1. Initial program 52.5

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

    3. Applied add-sqr-sqrt_binary6452.5

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

    4. Applied difference-of-squares_binary6452.5

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

    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_binary6435.2

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

    7. Simplified35.2

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

    8. Simplified35.2

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

    10. Applied associate-*r/_binary6435.2

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

    11. Simplified35.2

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

    if 7.7959870460050154e157 < 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 add-sqr-sqrt_binary6464.0

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

    4. Applied difference-of-squares_binary6464.0

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

    5. Applied times-frac_binary6458.2

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

    6. Applied times-frac_binary6451.9

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

    7. Simplified51.9

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

    8. Simplified51.9

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

    10. Applied associate-*r/_binary6451.9

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

    11. Simplified51.9

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

    13. Applied *-un-lft-identity_binary6451.9

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

    14. Applied *-un-lft-identity_binary6451.9

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

    15. Applied times-frac_binary6451.9

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

    16. Applied times-frac_binary6451.9

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

    17. Applied associate-*r*_binary6451.9

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

    18. Simplified51.9

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

    19. Taylor expanded around inf 36.9

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

  4. Final simplification35.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.795987046005015 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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)))