Average Error: 16.1 → 6.0
Time: 4.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 87164043717896432:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} - 1\right) + \mathsf{fma}\left(-1, 1, 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 87164043717896432:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} - 1\right) + \mathsf{fma}\left(-1, 1, 1\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\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) {
	return ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 8.716404371789643e+16)) {
		VAR = (((beta / ((alpha + beta) + 2.0)) - ((((cbrt((alpha / ((alpha + beta) + 2.0))) * cbrt((alpha / ((alpha + beta) + 2.0)))) * cbrt((alpha / ((alpha + beta) + 2.0)))) - 1.0) + fma(-1.0, 1.0, 1.0))) / 2.0);
	} else {
		VAR = (((beta / ((alpha + beta) + 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 8.716404371789643e+16

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.4

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

    6. Applied add-cube-cbrt0.4

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

    7. Applied add-cube-cbrt0.4

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

    8. Applied prod-diff0.4

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

    9. Simplified0.4

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

    10. Simplified0.4

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

    if 8.716404371789643e+16 < alpha

    1. Initial program 50.9

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

    3. Applied div-sub50.9

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

    4. Applied associate-+l-49.2

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

    5. Taylor expanded around inf 18.3

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

    6. Simplified18.3

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 87164043717896432:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} - 1\right) + \mathsf{fma}\left(-1, 1, 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))