Average Error: 16.4 → 16.4
Time: 20.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\frac{e^{\log \left(\frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}, {1}^{6}\right)}}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, 1 \cdot 1\right)}\right)}}{2}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\frac{e^{\log \left(\frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}, {1}^{6}\right)}}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, 1 \cdot 1\right)}\right)}}{2}
double f(double alpha, double beta) {
        double r64471 = beta;
        double r64472 = alpha;
        double r64473 = r64471 - r64472;
        double r64474 = r64472 + r64471;
        double r64475 = 2.0;
        double r64476 = r64474 + r64475;
        double r64477 = r64473 / r64476;
        double r64478 = 1.0;
        double r64479 = r64477 + r64478;
        double r64480 = r64479 / r64475;
        return r64480;
}
↓
double f(double alpha, double beta) {
        double r64481 = 1.0;
        double r64482 = 3.0;
        double r64483 = pow(r64481, r64482);
        double r64484 = pow(r64483, r64482);
        double r64485 = beta;
        double r64486 = alpha;
        double r64487 = r64485 - r64486;
        double r64488 = r64486 + r64485;
        double r64489 = 2.0;
        double r64490 = r64488 + r64489;
        double r64491 = r64487 / r64490;
        double r64492 = pow(r64491, r64482);
        double r64493 = pow(r64492, r64482);
        double r64494 = r64484 + r64493;
        double r64495 = r64492 - r64483;
        double r64496 = 6.0;
        double r64497 = pow(r64481, r64496);
        double r64498 = fma(r64495, r64492, r64497);
        double r64499 = r64494 / r64498;
        double r64500 = r64491 - r64481;
        double r64501 = r64481 * r64481;
        double r64502 = fma(r64500, r64491, r64501);
        double r64503 = r64499 / r64502;
        double r64504 = log(r64503);
        double r64505 = exp(r64504);
        double r64506 = r64505 / r64489;
        return r64506;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 16.4

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

  2. Simplified16.4

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

  4. Applied add-exp-log16.4

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

  5. Simplified16.4

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

  7. Applied flip3-+16.4

    \[\leadsto \frac{e^{\log \color{blue}{\left(\frac{{1}^{3} + {\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}}{1 \cdot 1 + \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} - 1 \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}\right)}}}{2}\]

  8. Simplified16.4

    \[\leadsto \frac{e^{\log \left(\frac{\color{blue}{{1}^{3} + {\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}}}{1 \cdot 1 + \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} - 1 \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}\right)}}{2}\]

  9. Simplified16.4

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

  11. Applied flip3-+16.4

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

  12. Simplified16.4

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

  13. Simplified16.4

    \[\leadsto \frac{e^{\log \left(\frac{\frac{{\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}\right)}^{3} + {\left({1}^{3}\right)}^{3}}{\color{blue}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}, {1}^{6}\right)}}}{\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} - 1, \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, 1 \cdot 1\right)}\right)}}{2}\]

  14. Final simplification16.4

    \[\leadsto \frac{e^{\log \left(\frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}, {1}^{6}\right)}}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, 1 \cdot 1\right)}\right)}}{2}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))