Average Error: 24.1 → 12.5
Time: 12.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1\right)}^{3}}}{2}\]
\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(\left(\alpha + \beta\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1\right)}^{3}}}{2}
double f(double alpha, double beta, double i) {
        double r157036 = alpha;
        double r157037 = beta;
        double r157038 = r157036 + r157037;
        double r157039 = r157037 - r157036;
        double r157040 = r157038 * r157039;
        double r157041 = 2.0;
        double r157042 = i;
        double r157043 = r157041 * r157042;
        double r157044 = r157038 + r157043;
        double r157045 = r157040 / r157044;
        double r157046 = r157044 + r157041;
        double r157047 = r157045 / r157046;
        double r157048 = 1.0;
        double r157049 = r157047 + r157048;
        double r157050 = r157049 / r157041;
        return r157050;
}
↓
double f(double alpha, double beta, double i) {
        double r157051 = alpha;
        double r157052 = beta;
        double r157053 = r157051 + r157052;
        double r157054 = 1.0;
        double r157055 = 2.0;
        double r157056 = i;
        double r157057 = r157055 * r157056;
        double r157058 = r157053 + r157057;
        double r157059 = r157058 + r157055;
        double r157060 = sqrt(r157059);
        double r157061 = r157054 / r157060;
        double r157062 = r157052 - r157051;
        double r157063 = r157062 / r157058;
        double r157064 = r157063 / r157060;
        double r157065 = r157061 * r157064;
        double r157066 = r157053 * r157065;
        double r157067 = 1.0;
        double r157068 = r157066 + r157067;
        double r157069 = 3.0;
        double r157070 = pow(r157068, r157069);
        double r157071 = cbrt(r157070);
        double r157072 = r157071 / r157055;
        return r157072;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 24.1

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

  3. Applied *-un-lft-identity24.1

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

  4. Applied *-un-lft-identity24.1

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

  5. Applied times-frac12.5

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

  6. Applied times-frac12.4

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

  7. Simplified12.4

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

  9. Applied add-cbrt-cube12.5

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

  10. Simplified12.5

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

  12. Applied add-sqr-sqrt12.5

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

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

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

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

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

  15. Applied times-frac12.5

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

  16. Applied times-frac12.5

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

  17. Simplified12.5

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

  18. Final simplification12.5

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

Reproduce

herbie shell --seed 2019356 
(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))