Average Error: 23.9 → 12.1
Time: 34.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}\right)}{2}\]
\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{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}\right)}{2}
double f(double alpha, double beta, double i) {
        double r92933 = alpha;
        double r92934 = beta;
        double r92935 = r92933 + r92934;
        double r92936 = r92934 - r92933;
        double r92937 = r92935 * r92936;
        double r92938 = 2.0;
        double r92939 = i;
        double r92940 = r92938 * r92939;
        double r92941 = r92935 + r92940;
        double r92942 = r92937 / r92941;
        double r92943 = r92941 + r92938;
        double r92944 = r92942 / r92943;
        double r92945 = 1.0;
        double r92946 = r92944 + r92945;
        double r92947 = r92946 / r92938;
        return r92947;
}
↓
double f(double alpha, double beta, double i) {
        double r92948 = beta;
        double r92949 = alpha;
        double r92950 = r92948 - r92949;
        double r92951 = 2.0;
        double r92952 = i;
        double r92953 = r92949 + r92948;
        double r92954 = fma(r92951, r92952, r92953);
        double r92955 = r92954 + r92951;
        double r92956 = r92950 / r92955;
        double r92957 = r92953 / r92954;
        double r92958 = 1.0;
        double r92959 = fma(r92956, r92957, r92958);
        double r92960 = exp(r92959);
        double r92961 = log(r92960);
        double r92962 = r92961 / r92951;
        return r92962;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

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

  2. Simplified12.1

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

  4. Applied add-log-exp12.1

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

  5. Final simplification12.1

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

Reproduce

herbie shell --seed 2019323 +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))