Average Error: 24.2 → 11.5
Time: 5.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.3286644476139376 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{1}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 7.3286644476139376 \cdot 10^{139}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{1}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\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 <= 7.328664447613938e+139)) {
		VAR = ((((((alpha + beta) / fma(i, 2.0, (alpha + beta))) / (1.0 / (beta - alpha))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0);
	} else {
		VAR = (fma(2.0, (1.0 / alpha), ((8.0 * (1.0 / pow(alpha, 3.0))) - (4.0 * (1.0 / pow(alpha, 2.0))))) / 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 < 7.328664447613938e+139

    1. Initial program 15.6

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

    3. Applied associate-/l*4.8

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

    5. Applied div-inv4.8

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

    6. Applied associate-/r*4.8

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

    7. Simplified4.8

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

    if 7.328664447613938e+139 < alpha

    1. Initial program 62.5

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

    3. Applied associate-/l*46.8

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

    5. Applied div-inv47.0

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

    6. Applied associate-/r*46.9

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

    7. Simplified46.9

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

    9. Applied div-inv46.8

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

    10. Applied fma-def47.5

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

    11. Taylor expanded around inf 41.9

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

    12. Simplified41.9

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.3286644476139376 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{1}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))