Average Error: 24.5 → 12.7
Time: 21.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\mathsf{fma}\left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}, \alpha + \beta, 1\right)}{2}\]
\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} + 1}{2}
↓
\frac{\mathsf{fma}\left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{2 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt[3]{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}, \alpha + \beta, 1\right)}{2}
double f(double alpha, double beta, double i) {
        double r116967 = alpha;
        double r116968 = beta;
        double r116969 = r116967 + r116968;
        double r116970 = r116968 - r116967;
        double r116971 = r116969 * r116970;
        double r116972 = 2.0;
        double r116973 = i;
        double r116974 = r116972 * r116973;
        double r116975 = r116969 + r116974;
        double r116976 = r116971 / r116975;
        double r116977 = r116975 + r116972;
        double r116978 = r116976 / r116977;
        double r116979 = 1.0;
        double r116980 = r116978 + r116979;
        double r116981 = r116980 / r116972;
        return r116981;
}
↓
double f(double alpha, double beta, double i) {
        double r116982 = beta;
        double r116983 = alpha;
        double r116984 = r116982 - r116983;
        double r116985 = cbrt(r116984);
        double r116986 = i;
        double r116987 = 2.0;
        double r116988 = r116983 + r116982;
        double r116989 = fma(r116986, r116987, r116988);
        double r116990 = cbrt(r116989);
        double r116991 = r116985 / r116990;
        double r116992 = r116991 * r116991;
        double r116993 = r116987 + r116989;
        double r116994 = r116985 / r116993;
        double r116995 = r116994 / r116990;
        double r116996 = r116992 * r116995;
        double r116997 = 1.0;
        double r116998 = fma(r116996, r116988, r116997);
        double r116999 = r116998 / r116987;
        return r116999;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.5

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

  2. Simplified12.7

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

  4. Applied *-un-lft-identity12.7

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

  5. Applied add-cube-cbrt12.9

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

  6. Applied add-cube-cbrt12.7

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

  7. Applied times-frac12.7

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

  8. Applied times-frac12.7

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

  9. Simplified12.7

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

  10. Simplified12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))