Average Error: 23.2 → 11.9
Time: 14.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \log \left(e^{\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right), 1\right)}{2}\]
\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(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \log \left(e^{\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right), 1\right)}{2}
double f(double alpha, double beta, double i) {
        double r70082 = alpha;
        double r70083 = beta;
        double r70084 = r70082 + r70083;
        double r70085 = r70083 - r70082;
        double r70086 = r70084 * r70085;
        double r70087 = 2.0;
        double r70088 = i;
        double r70089 = r70087 * r70088;
        double r70090 = r70084 + r70089;
        double r70091 = r70086 / r70090;
        double r70092 = r70090 + r70087;
        double r70093 = r70091 / r70092;
        double r70094 = 1.0;
        double r70095 = r70093 + r70094;
        double r70096 = r70095 / r70087;
        return r70096;
}
↓
double f(double alpha, double beta, double i) {
        double r70097 = beta;
        double r70098 = alpha;
        double r70099 = r70097 - r70098;
        double r70100 = 2.0;
        double r70101 = i;
        double r70102 = r70098 + r70097;
        double r70103 = fma(r70100, r70101, r70102);
        double r70104 = r70103 + r70100;
        double r70105 = r70099 / r70104;
        double r70106 = r70102 / r70103;
        double r70107 = exp(r70106);
        double r70108 = log(r70107);
        double r70109 = 1.0;
        double r70110 = fma(r70105, r70108, r70109);
        double r70111 = r70110 / r70100;
        return r70111;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.2

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

  2. Simplified11.9

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

  4. Applied add-log-exp11.9

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

  6. Applied add-log-exp11.9

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

  8. Applied rem-log-exp11.9

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

  9. Final simplification11.9

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

Reproduce

herbie shell --seed 2019235 +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))