Average Error: 54.0 → 36.4
Time: 12.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1.3297226358063274 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;i \le 1.3297226358063274 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
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 VAR;
	if ((i <= 1.3297226358063274e+154)) {
		VAR = (((1.0 / (((alpha + beta) + (2.0 * i)) + sqrt(1.0))) * ((i * ((alpha + beta) + i)) / (fma(i, 2.0, (alpha + beta)) - sqrt(1.0)))) / (fma(i, 2.0, (alpha + beta)) / (fma(beta, alpha, (i * ((alpha + beta) + i))) / fma(i, 2.0, (alpha + beta)))));
	} else {
		VAR = 0.0;
	}
	return VAR;
}

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 < 1.3297226358063274e+154

    1. Initial program 44.2

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

    2. Simplified41.1

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

    4. Applied associate-/l*33.2

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

    6. Applied *-un-lft-identity33.2

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

    7. Applied *-un-lft-identity33.2

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

    8. Applied times-frac33.2

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

    9. Applied times-frac18.3

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

    10. Applied associate-/r*15.7

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

    11. Simplified15.6

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

    13. Applied *-un-lft-identity15.6

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

    14. Applied add-sqr-sqrt15.6

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

    15. Applied difference-of-squares15.6

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

    16. Applied times-frac11.4

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

    17. Applied *-un-lft-identity11.4

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

    18. Applied times-frac11.5

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

    19. Simplified11.5

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

    20. Simplified11.5

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

    if 1.3297226358063274e+154 < i

    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. Simplified64.0

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

    3. Taylor expanded around inf 61.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification36.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.3297226358063274 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))