Average Error: 53.8 → 38.8
Time: 9.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 2.96611707754627387 \cdot 10^{104}:\\ \;\;\;\;\frac{i}{\left(-1\right) + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\left(-1\right) + \mathsf{fma}\left(\mathsf{fma}\left(i, i, \alpha \cdot i\right), 4, {\alpha}^{2}\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 0.25\\ \end{array}\]
\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 2.96611707754627387 \cdot 10^{104}:\\
\;\;\;\;\frac{i}{\left(-1\right) + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\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)}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{i}{\left(-1\right) + \mathsf{fma}\left(\mathsf{fma}\left(i, i, \alpha \cdot i\right), 4, {\alpha}^{2}\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 0.25\\

\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 <= 2.966117077546274e+104)) {
		VAR = ((i / (-1.0 + (fma(i, 2.0, (alpha + beta)) * fma(i, 2.0, (alpha + beta))))) * (((alpha + beta) + i) / ((fma(i, 2.0, (alpha + beta)) * fma(i, 2.0, (alpha + beta))) / fma(beta, alpha, (i * ((alpha + beta) + i))))));
	} else {
		VAR = (((i / (-1.0 + fma(fma(i, i, (alpha * i)), 4.0, pow(alpha, 2.0)))) * ((alpha + beta) + i)) * 0.25);
	}
	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 < 2.966117077546274e+104

    1. Initial program 34.2

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

    2. Simplified31.0

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

    4. Applied associate-*l*30.9

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

    5. Applied associate-/l*17.2

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

    6. Applied associate-/r/14.0

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

    7. Simplified14.1

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

    9. Applied associate-*l*14.1

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

    10. Simplified14.1

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

    12. Applied clear-num14.1

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

    13. Applied associate-*l/14.1

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

    14. Simplified14.1

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

    if 2.966117077546274e+104 < 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. Simplified63.6

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

    4. Applied associate-*l*63.6

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

    5. Applied associate-/l*53.1

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

    6. Applied associate-/r/52.8

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

    7. Simplified52.8

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

    8. Taylor expanded around 0 51.3

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

    9. Taylor expanded around 0 51.7

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

    10. Simplified51.7

      \[\leadsto \left(\frac{i}{\left(-1\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(i, i, \alpha \cdot i\right), 4, {\alpha}^{2}\right)}} \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 0.25\]
  3. Recombined 2 regimes into one program.

  4. Final simplification38.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.96611707754627387 \cdot 10^{104}:\\ \;\;\;\;\frac{i}{\left(-1\right) + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\left(-1\right) + \mathsf{fma}\left(\mathsf{fma}\left(i, i, \alpha \cdot i\right), 4, {\alpha}^{2}\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 0.25\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))