Octave 3.8, jcobi/2

Average Error: 23.7 → 11.9

Time: 38.5s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r93477 = alpha;
        double r93478 = beta;
        double r93479 = r93477 + r93478;
        double r93480 = r93478 - r93477;
        double r93481 = r93479 * r93480;
        double r93482 = 2.0;
        double r93483 = i;
        double r93484 = r93482 * r93483;
        double r93485 = r93479 + r93484;
        double r93486 = r93481 / r93485;
        double r93487 = r93485 + r93482;
        double r93488 = r93486 / r93487;
        double r93489 = 1.0;
        double r93490 = r93488 + r93489;
        double r93491 = r93490 / r93482;
        return r93491;
}

↓

double f(double alpha, double beta, double i) {
        double r93492 = beta;
        double r93493 = alpha;
        double r93494 = r93492 - r93493;
        double r93495 = 2.0;
        double r93496 = i;
        double r93497 = r93493 + r93492;
        double r93498 = fma(r93495, r93496, r93497);
        double r93499 = r93498 + r93495;
        double r93500 = r93494 / r93499;
        double r93501 = r93497 / r93498;
        double r93502 = r93500 * r93501;
        double r93503 = r93502 * r93502;
        double r93504 = r93503 * r93502;
        double r93505 = 1.0;
        double r93506 = 3.0;
        double r93507 = pow(r93505, r93506);
        double r93508 = r93504 + r93507;
        double r93509 = r93505 - r93502;
        double r93510 = r93505 * r93509;
        double r93511 = fma(r93502, r93502, r93510);
        double r93512 = r93508 / r93511;
        double r93513 = r93512 / r93495;
        return r93513;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 23.7

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

2. Simplified 11.9

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

4. Applied fma-udef 11.9

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

6. Applied flip3-+ 11.9

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

7. Simplified 11.9

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

9. Applied add-cbrt-cube 11.9

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

10. Applied rem-cube-cbrt 11.9

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

11. Final simplification 11.9

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

Reproduce

herbie shell --seed 2019209 +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))