Average Error: 23.3 → 11.5
Time: 12.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.755371486194466 \cdot 10^{248}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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 1.755371486194466 \cdot 10^{248}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r104547 = alpha;
        double r104548 = beta;
        double r104549 = r104547 + r104548;
        double r104550 = r104548 - r104547;
        double r104551 = r104549 * r104550;
        double r104552 = 2.0;
        double r104553 = i;
        double r104554 = r104552 * r104553;
        double r104555 = r104549 + r104554;
        double r104556 = r104551 / r104555;
        double r104557 = r104555 + r104552;
        double r104558 = r104556 / r104557;
        double r104559 = 1.0;
        double r104560 = r104558 + r104559;
        double r104561 = r104560 / r104552;
        return r104561;
}
↓
double f(double alpha, double beta, double i) {
        double r104562 = alpha;
        double r104563 = 1.7553714861944655e+248;
        bool r104564 = r104562 <= r104563;
        double r104565 = beta;
        double r104566 = r104562 + r104565;
        double r104567 = 2.0;
        double r104568 = i;
        double r104569 = fma(r104567, r104568, r104566);
        double r104570 = r104569 + r104567;
        double r104571 = sqrt(r104570);
        double r104572 = cbrt(r104571);
        double r104573 = r104572 * r104572;
        double r104574 = r104566 / r104573;
        double r104575 = cbrt(r104570);
        double r104576 = r104575 * r104575;
        double r104577 = r104574 / r104576;
        double r104578 = r104565 - r104562;
        double r104579 = r104578 / r104569;
        double r104580 = 1.0;
        double r104581 = fma(r104577, r104579, r104580);
        double r104582 = 3.0;
        double r104583 = pow(r104581, r104582);
        double r104584 = cbrt(r104583);
        double r104585 = r104584 / r104567;
        double r104586 = 8.0;
        double r104587 = pow(r104562, r104582);
        double r104588 = r104586 / r104587;
        double r104589 = r104567 / r104562;
        double r104590 = 4.0;
        double r104591 = r104562 * r104562;
        double r104592 = r104590 / r104591;
        double r104593 = r104589 - r104592;
        double r104594 = r104588 + r104593;
        double r104595 = r104594 / r104567;
        double r104596 = r104564 ? r104585 : r104595;
        return r104596;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.7553714861944655e+248

    1. Initial program 20.6

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

    2. Simplified9.5

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

    4. Applied add-cube-cbrt9.6

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

    5. Applied *-un-lft-identity9.6

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

    6. Applied times-frac9.6

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

    8. Applied add-sqr-sqrt9.6

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

    9. Applied cbrt-prod9.6

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

    11. Applied add-cbrt-cube9.5

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

    12. Simplified9.6

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

    if 1.7553714861944655e+248 < alpha

    1. Initial program 64.0

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

    2. Simplified54.3

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

    3. Taylor expanded around inf 40.5

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

    4. Simplified40.5

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.755371486194466 \cdot 10^{248}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))