Average Error: 24.6 → 11.5
Time: 8.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.026667996136292 \cdot 10^{102}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 2\right)\right)}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 4.026667996136292 \cdot 10^{102}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 2\right)\right)}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}, 1\right)\right)}^{3}}}{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 temp;
	if ((alpha <= 4.026667996136292e+102)) {
		temp = (cbrt(pow(fma((alpha + beta), (((cbrt((beta - alpha)) * cbrt((beta - alpha))) / (cbrt(((alpha + beta) + (2.0 * i))) * cbrt(((alpha + beta) + (2.0 * i))))) * ((cbrt((beta - alpha)) / (beta + (alpha + fma(i, 2.0, 2.0)))) / cbrt(((alpha + beta) + (2.0 * i))))), 1.0), 3.0)) / 2.0);
	} else {
		temp = (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 temp;
}

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 < 4.026667996136292e+102

    1. Initial program 14.5

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

    3. Applied *-un-lft-identity14.5

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

    4. Applied *-un-lft-identity14.5

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

    5. Applied times-frac3.2

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

    6. Applied times-frac3.1

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

    7. Applied fma-def3.1

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

    9. Applied add-cbrt-cube3.1

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

    10. Simplified3.1

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

    12. Applied *-un-lft-identity3.1

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

    13. Applied add-cube-cbrt3.2

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

    14. Applied add-cube-cbrt3.2

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

    15. Applied times-frac3.2

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

    16. Applied times-frac3.2

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

    17. Simplified3.2

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

    18. Simplified3.2

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

    if 4.026667996136292e+102 < alpha

    1. Initial program 59.8

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

    2. Taylor expanded around inf 40.3

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

    3. Simplified40.3

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.026667996136292 \cdot 10^{102}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\alpha + \mathsf{fma}\left(i, 2, 2\right)\right)}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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))