Octave 3.8, jcobi/2

Average Error: 24.4 → 12.1

Time: 21.5s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r86323 = alpha;
        double r86324 = beta;
        double r86325 = r86323 + r86324;
        double r86326 = r86324 - r86323;
        double r86327 = r86325 * r86326;
        double r86328 = 2.0;
        double r86329 = i;
        double r86330 = r86328 * r86329;
        double r86331 = r86325 + r86330;
        double r86332 = r86327 / r86331;
        double r86333 = r86331 + r86328;
        double r86334 = r86332 / r86333;
        double r86335 = 1.0;
        double r86336 = r86334 + r86335;
        double r86337 = r86336 / r86328;
        return r86337;
}

↓

double f(double alpha, double beta, double i) {
        double r86338 = 1.0;
        double r86339 = 2.0;
        double r86340 = i;
        double r86341 = alpha;
        double r86342 = beta;
        double r86343 = r86341 + r86342;
        double r86344 = fma(r86339, r86340, r86343);
        double r86345 = r86344 + r86339;
        double r86346 = sqrt(r86345);
        double r86347 = r86338 / r86346;
        double r86348 = sqrt(r86347);
        double r86349 = cbrt(r86345);
        double r86350 = fabs(r86349);
        double r86351 = sqrt(r86350);
        double r86352 = r86348 / r86351;
        double r86353 = sqrt(r86349);
        double r86354 = sqrt(r86353);
        double r86355 = r86348 / r86354;
        double r86356 = r86352 * r86355;
        double r86357 = r86342 - r86341;
        double r86358 = sqrt(r86346);
        double r86359 = r86357 / r86358;
        double r86360 = r86356 * r86359;
        double r86361 = r86343 / r86344;
        double r86362 = 1.0;
        double r86363 = fma(r86360, r86361, r86362);
        double r86364 = r86363 / r86339;
        return r86364;
}

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. Simplified 12.0

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

4. Applied add-sqr-sqrt 12.1

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

5. Applied *-un-lft-identity 12.1

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

6. Applied times-frac 12.1

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

8. Applied add-sqr-sqrt 12.1

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

9. Applied sqrt-prod 12.1

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

10. Applied *-un-lft-identity 12.1

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

11. Applied times-frac 12.1

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

12. Applied associate-*r* 12.1

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

13. Simplified 12.1

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

15. Applied add-cube-cbrt 12.1

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

16. Applied sqrt-prod 12.1

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

17. Applied sqrt-prod 12.1

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

18. Applied add-sqr-sqrt 12.1

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

19. Applied times-frac 12.1

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

20. Simplified 12.1

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

21. Final simplification 12.1

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))