Average Error: 24.4 → 12.1
Time: 24.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
\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}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}
double f(double alpha, double beta, double i) {
        double r80343 = alpha;
        double r80344 = beta;
        double r80345 = r80343 + r80344;
        double r80346 = r80344 - r80343;
        double r80347 = r80345 * r80346;
        double r80348 = 2.0;
        double r80349 = i;
        double r80350 = r80348 * r80349;
        double r80351 = r80345 + r80350;
        double r80352 = r80347 / r80351;
        double r80353 = r80351 + r80348;
        double r80354 = r80352 / r80353;
        double r80355 = 1.0;
        double r80356 = r80354 + r80355;
        double r80357 = r80356 / r80348;
        return r80357;
}
↓
double f(double alpha, double beta, double i) {
        double r80358 = beta;
        double r80359 = alpha;
        double r80360 = r80358 - r80359;
        double r80361 = 2.0;
        double r80362 = i;
        double r80363 = r80359 + r80358;
        double r80364 = fma(r80361, r80362, r80363);
        double r80365 = r80364 + r80361;
        double r80366 = r80360 / r80365;
        double r80367 = r80363 / r80364;
        double r80368 = 1.0;
        double r80369 = fma(r80366, r80367, r80368);
        double r80370 = r80369 / r80361;
        return r80370;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.4

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

  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. Final simplification12.1

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

Reproduce

herbie shell --seed 2019303 +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))