Average Error: 24.2 → 12.6
Time: 24.0s
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 r82226 = alpha;
        double r82227 = beta;
        double r82228 = r82226 + r82227;
        double r82229 = r82227 - r82226;
        double r82230 = r82228 * r82229;
        double r82231 = 2.0;
        double r82232 = i;
        double r82233 = r82231 * r82232;
        double r82234 = r82228 + r82233;
        double r82235 = r82230 / r82234;
        double r82236 = r82234 + r82231;
        double r82237 = r82235 / r82236;
        double r82238 = 1.0;
        double r82239 = r82237 + r82238;
        double r82240 = r82239 / r82231;
        return r82240;
}
↓
double f(double alpha, double beta, double i) {
        double r82241 = alpha;
        double r82242 = beta;
        double r82243 = r82241 + r82242;
        double r82244 = 2.0;
        double r82245 = i;
        double r82246 = fma(r82244, r82245, r82243);
        double r82247 = r82243 / r82246;
        double r82248 = r82242 - r82241;
        double r82249 = r82246 + r82244;
        double r82250 = cbrt(r82249);
        double r82251 = fabs(r82250);
        double r82252 = r82248 / r82251;
        double r82253 = sqrt(r82250);
        double r82254 = r82252 / r82253;
        double r82255 = sqrt(r82249);
        double r82256 = r82254 / r82255;
        double r82257 = 1.0;
        double r82258 = fma(r82247, r82256, r82257);
        double r82259 = 3.0;
        double r82260 = pow(r82258, r82259);
        double r82261 = cbrt(r82260);
        double r82262 = r82261 / r82244;
        return r82262;
}

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))