Average Error: 16.2 → 6.7
Time: 6.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.8710423887153736 \cdot 10^{42}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right), -\left(\left(\alpha + \beta\right) + 2\right) \cdot \mathsf{fma}\left({\left(\frac{\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)}^{3}, {\left(\frac{\sqrt[3]{\alpha}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)}^{3}, -{1}^{3}\right)\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9.8710423887153736 \cdot 10^{42}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right), -\left(\left(\alpha + \beta\right) + 2\right) \cdot \mathsf{fma}\left({\left(\frac{\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)}^{3}, {\left(\frac{\sqrt[3]{\alpha}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)}^{3}, -{1}^{3}\right)\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\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 <= 9.871042388715374e+42)) {
		VAR = ((fma(beta, fma((alpha / ((alpha + beta) + 2.0)), (alpha / ((alpha + beta) + 2.0)), fma(1.0, 1.0, ((alpha / ((alpha + beta) + 2.0)) * 1.0))), -(((alpha + beta) + 2.0) * fma(pow(((cbrt(alpha) * cbrt(alpha)) / sqrt(((alpha + beta) + 2.0))), 3.0), pow((cbrt(alpha) / sqrt(((alpha + beta) + 2.0))), 3.0), -pow(1.0, 3.0)))) / (fma((alpha / ((alpha + beta) + 2.0)), (alpha / ((alpha + beta) + 2.0)), fma(1.0, 1.0, ((alpha / ((alpha + beta) + 2.0)) * 1.0))) * ((alpha + beta) + 2.0))) / 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 < 9.871042388715374e+42

    1. Initial program 2.1

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

    3. Applied div-sub2.0

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

    4. Applied associate-+l-2.0

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

    6. Applied flip3--2.0

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

    7. Applied frac-sub2.1

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

    8. Simplified2.1

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

    9. Simplified2.1

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

    11. Applied add-sqr-sqrt2.1

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

    12. Applied add-cube-cbrt2.1

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

    13. Applied times-frac2.1

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

    14. Applied unpow-prod-down2.1

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

    15. Applied fma-neg2.1

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

    if 9.871042388715374e+42 < alpha

    1. Initial program 51.5

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

    3. Applied div-sub51.5

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

    4. Applied associate-+l-49.8

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

    5. Taylor expanded around inf 18.0

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

    6. Simplified18.0

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.8710423887153736 \cdot 10^{42}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right), -\left(\left(\alpha + \beta\right) + 2\right) \cdot \mathsf{fma}\left({\left(\frac{\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)}^{3}, {\left(\frac{\sqrt[3]{\alpha}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)}^{3}, -{1}^{3}\right)\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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))