Average Error: 23.9 → 11.1
Time: 29.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.959262384410508 \cdot 10^{+188}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, \beta - \alpha, 1.0\right)\right)\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.959262384410508 \cdot 10^{+188}:\\
\;\;\;\;\frac{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, \beta - \alpha, 1.0\right)\right)\right)\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r4513027 = alpha;
        double r4513028 = beta;
        double r4513029 = r4513027 + r4513028;
        double r4513030 = r4513028 - r4513027;
        double r4513031 = r4513029 * r4513030;
        double r4513032 = 2.0;
        double r4513033 = i;
        double r4513034 = r4513032 * r4513033;
        double r4513035 = r4513029 + r4513034;
        double r4513036 = r4513031 / r4513035;
        double r4513037 = 2.0;
        double r4513038 = r4513035 + r4513037;
        double r4513039 = r4513036 / r4513038;
        double r4513040 = 1.0;
        double r4513041 = r4513039 + r4513040;
        double r4513042 = r4513041 / r4513037;
        return r4513042;
}
↓
double f(double alpha, double beta, double i) {
        double r4513043 = alpha;
        double r4513044 = 2.959262384410508e+188;
        bool r4513045 = r4513043 <= r4513044;
        double r4513046 = beta;
        double r4513047 = r4513046 + r4513043;
        double r4513048 = 2.0;
        double r4513049 = i;
        double r4513050 = fma(r4513048, r4513049, r4513047);
        double r4513051 = r4513047 / r4513050;
        double r4513052 = 2.0;
        double r4513053 = r4513052 + r4513050;
        double r4513054 = r4513051 / r4513053;
        double r4513055 = r4513046 - r4513043;
        double r4513056 = 1.0;
        double r4513057 = fma(r4513054, r4513055, r4513056);
        double r4513058 = log(r4513057);
        double r4513059 = expm1(r4513058);
        double r4513060 = log1p(r4513059);
        double r4513061 = exp(r4513060);
        double r4513062 = r4513061 / r4513052;
        double r4513063 = r4513052 / r4513043;
        double r4513064 = 4.0;
        double r4513065 = r4513043 * r4513043;
        double r4513066 = r4513064 / r4513065;
        double r4513067 = r4513063 - r4513066;
        double r4513068 = 8.0;
        double r4513069 = r4513068 / r4513043;
        double r4513070 = r4513069 / r4513065;
        double r4513071 = r4513067 + r4513070;
        double r4513072 = r4513071 / r4513052;
        double r4513073 = r4513045 ? r4513062 : r4513072;
        return r4513073;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.959262384410508e+188

    1. Initial program 17.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Simplified17.9

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0}{2.0}}\]
    3. Using strategy rm

    4. Applied add-exp-log17.9

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0\right)}}}{2.0}\]

    5. Simplified14.7

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) \cdot \left(2.0 + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right)}, \beta - \alpha, 1.0\right)\right)}}}{2.0}\]
    6. Using strategy rm

    7. Applied associate-/r*6.7

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2.0 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}, \beta - \alpha, 1.0\right)\right)}}{2.0}\]
    8. Using strategy rm

    9. Applied log1p-expm1-u6.8

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2.0 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}, \beta - \alpha, 1.0\right)\right)\right)\right)}}}{2.0}\]

    if 2.959262384410508e+188 < alpha

    1. Initial program 63.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Simplified62.5

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0}{2.0}}\]
    3. Using strategy rm

    4. Applied add-exp-log62.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0\right)}}}{2.0}\]

    5. Simplified54.0

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) \cdot \left(2.0 + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right)}, \beta - \alpha, 1.0\right)\right)}}}{2.0}\]
    6. Using strategy rm

    7. Applied associate-/r*49.8

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2.0 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}, \beta - \alpha, 1.0\right)\right)}}{2.0}\]

    8. Taylor expanded around inf 39.9

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    9. Simplified39.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.959262384410508 \cdot 10^{+188}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, \beta - \alpha, 1.0\right)\right)\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))