Average Error: 24.3 → 11.3
Time: 5.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.26835559615667094 \cdot 10^{86}:\\ \;\;\;\;\frac{\frac{\frac{1}{-\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}} \cdot \left(-\left(\alpha + \beta\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.26835559615667094 \cdot 10^{86}:\\
\;\;\;\;\frac{\frac{\frac{1}{-\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}} \cdot \left(-\left(\alpha + \beta\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double code(double alpha, double beta, double i) {
	return ((((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 1.268355596156671e+86)) {
		VAR = (((((1.0 / -(fma(i, 2.0, (alpha + beta)) / beta)) * -(alpha + beta)) / (((alpha + beta) + (2.0 * i)) + 2.0)) - (((alpha / (((alpha + beta) + (2.0 * i)) / (alpha + beta))) / (((alpha + beta) + (2.0 * i)) + 2.0)) - 1.0)) / 2.0);
	} else {
		VAR = ((((1.0 / ((fma(i, 2.0, (alpha + beta)) / beta) / (alpha + beta))) / (((alpha + beta) + (2.0 * i)) + 2.0)) - fma(4.0, (1.0 / pow(alpha, 2.0)), -fma(2.0, (1.0 / alpha), (8.0 * (1.0 / pow(alpha, 3.0)))))) / 2.0);
	}
	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 alpha < 1.268355596156671e+86

    1. Initial program 13.4

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

    3. Applied *-commutative13.4

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

    4. Applied associate-/l*2.6

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

    6. Applied div-sub2.6

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

    7. Applied div-sub2.6

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

    8. Applied associate-+l-2.5

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

    10. Applied clear-num2.5

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

    11. Simplified2.5

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

    13. Applied frac-2neg2.5

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

    14. Applied associate-/r/2.5

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

    if 1.268355596156671e+86 < alpha

    1. Initial program 58.0

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

    3. Applied *-commutative58.0

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

    4. Applied associate-/l*44.2

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

    6. Applied div-sub44.2

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

    7. Applied div-sub44.2

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

    8. Applied associate-+l-42.9

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

    10. Applied clear-num42.9

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

    11. Simplified42.9

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

    12. Taylor expanded around inf 38.4

      \[\leadsto \frac{\frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    13. Simplified38.4

      \[\leadsto \frac{\frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \color{blue}{\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.26835559615667094 \cdot 10^{86}:\\ \;\;\;\;\frac{\frac{\frac{1}{-\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}} \cdot \left(-\left(\alpha + \beta\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta}}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))