Average Error: 24.2 → 12.6
Time: 25.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{\frac{\frac{\beta - \alpha}{\left|\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right|}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, 1\right)\right)}^{3}}}{2}\]
\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{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{\frac{\frac{\beta - \alpha}{\left|\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right|}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, 1\right)\right)}^{3}}}{2}
double f(double alpha, double beta, double i) {
        double r88376 = alpha;
        double r88377 = beta;
        double r88378 = r88376 + r88377;
        double r88379 = r88377 - r88376;
        double r88380 = r88378 * r88379;
        double r88381 = 2.0;
        double r88382 = i;
        double r88383 = r88381 * r88382;
        double r88384 = r88378 + r88383;
        double r88385 = r88380 / r88384;
        double r88386 = r88384 + r88381;
        double r88387 = r88385 / r88386;
        double r88388 = 1.0;
        double r88389 = r88387 + r88388;
        double r88390 = r88389 / r88381;
        return r88390;
}
↓
double f(double alpha, double beta, double i) {
        double r88391 = alpha;
        double r88392 = beta;
        double r88393 = r88391 + r88392;
        double r88394 = 2.0;
        double r88395 = i;
        double r88396 = fma(r88394, r88395, r88393);
        double r88397 = r88393 / r88396;
        double r88398 = r88392 - r88391;
        double r88399 = r88396 + r88394;
        double r88400 = cbrt(r88399);
        double r88401 = fabs(r88400);
        double r88402 = r88398 / r88401;
        double r88403 = sqrt(r88400);
        double r88404 = r88402 / r88403;
        double r88405 = sqrt(r88399);
        double r88406 = r88404 / r88405;
        double r88407 = 1.0;
        double r88408 = fma(r88397, r88406, r88407);
        double r88409 = 3.0;
        double r88410 = pow(r88408, r88409);
        double r88411 = cbrt(r88410);
        double r88412 = r88411 / r88394;
        return r88412;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.2

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

  2. Simplified12.5

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

  4. Applied add-sqr-sqrt12.6

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

  5. Applied *-un-lft-identity12.6

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

  6. Applied times-frac12.6

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

  8. Applied add-sqr-sqrt12.6

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

  9. Applied associate-*l*12.6

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

  11. Applied add-cube-cbrt12.7

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

  12. Applied sqrt-prod12.7

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

  13. Applied associate-/r*12.6

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

  14. Simplified12.6

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

  16. Applied add-cbrt-cube12.6

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

  17. Simplified12.6

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

  18. Final simplification12.6

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

Reproduce

herbie shell --seed 2019326 +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))