Average Error: 53.8 → 36.6
Time: 11.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.0741625989722365 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\ \mathbf{elif}\;\beta \leq 2.0872971966169687 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;\beta \leq 1.5166568001122684 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 2.0741625989722365 \cdot 10^{-70}:\\
\;\;\;\;\frac{\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\

\mathbf{elif}\;\beta \leq 2.0872971966169687 \cdot 10^{+22}:\\
\;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;\beta \leq 1.5166568001122684 \cdot 10^{+231}:\\
\;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{\beta \cdot \alpha + 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}:\\
\;\;\;\;0\\

\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 2.0741625989722365e-70)
   (/
    (/
     (*
      (/
       (+ (* 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))
   (if (<= beta 2.0872971966169687e+22)
     (/
      (* (* i i) 0.25)
      (- (* (+ (+ beta alpha) (* i 2.0)) (+ (+ beta alpha) (* i 2.0))) 1.0))
     (if (<= beta 1.5166568001122684e+231)
       (/
        (*
         (/ (* i (+ i (+ beta alpha))) (+ (+ beta alpha) (* i 2.0)))
         (/
          (/
           (+ (* beta alpha) (* i (+ i (+ beta alpha))))
           (+ (+ beta alpha) (* i 2.0)))
          (- (+ (+ beta alpha) (* i 2.0)) 1.0)))
        (+ (+ (+ beta alpha) (* i 2.0)) 1.0))
       0.0))))
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 <= 2.0741625989722365e-70) {
		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 if (beta <= 2.0872971966169687e+22) {
		tmp = ((i * i) * 0.25) / ((((beta + alpha) + (i * 2.0)) * ((beta + alpha) + (i * 2.0))) - 1.0);
	} else if (beta <= 1.5166568001122684e+231) {
		tmp = (((i * (i + (beta + alpha))) / ((beta + alpha) + (i * 2.0))) * ((((beta * alpha) + (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 = 0.0;
	}
	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 4 regimes

  2. if beta < 2.0741625989722365e-70

    1. Initial program 50.8

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

    3. Applied difference-of-sqr-1_binary6450.8

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

    4. Applied times-frac_binary6435.4

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

    5. Applied times-frac_binary6434.3

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

    6. Simplified34.3

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

    7. Simplified34.3

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

    9. Applied associate-*l/_binary6434.3

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

    11. Applied associate-*r/_binary6434.3

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

    12. Simplified34.3

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

    14. Applied *-un-lft-identity_binary6434.3

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

    15. Applied times-frac_binary6434.3

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

    16. Simplified34.3

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

    if 2.0741625989722365e-70 < beta < 2.0872971966169687e22

    1. Initial program 51.7

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

    2. Taylor expanded around inf 37.0

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

    3. Simplified37.0

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

    if 2.0872971966169687e22 < beta < 1.51665680011226837e231

    1. Initial program 58.7

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

    3. Applied difference-of-sqr-1_binary6458.7

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

    4. Applied times-frac_binary6445.7

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

    5. Applied times-frac_binary6440.1

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

    6. Simplified40.1

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

    7. Simplified40.1

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

    9. Applied associate-*l/_binary6440.1

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

    if 1.51665680011226837e231 < 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. Taylor expanded around inf 43.0

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification36.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.0741625989722365 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(i \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\ \mathbf{elif}\;\beta \leq 2.0872971966169687 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;\beta \leq 1.5166568001122684 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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)))