Average Error: 23.9 → 12.1
Time: 1.2m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\log \left(e^{\frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + i \cdot 2}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + i \cdot 2}}}{\frac{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + i \cdot 2}}}} \cdot \left(\alpha + \beta\right) + 1}\right)}{2}\]
\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{\log \left(e^{\frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + i \cdot 2}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + i \cdot 2}}}{\frac{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + i \cdot 2}}}} \cdot \left(\alpha + \beta\right) + 1}\right)}{2}
double f(double alpha, double beta, double i) {
        double r5251075 = alpha;
        double r5251076 = beta;
        double r5251077 = r5251075 + r5251076;
        double r5251078 = r5251076 - r5251075;
        double r5251079 = r5251077 * r5251078;
        double r5251080 = 2.0;
        double r5251081 = i;
        double r5251082 = r5251080 * r5251081;
        double r5251083 = r5251077 + r5251082;
        double r5251084 = r5251079 / r5251083;
        double r5251085 = r5251083 + r5251080;
        double r5251086 = r5251084 / r5251085;
        double r5251087 = 1.0;
        double r5251088 = r5251086 + r5251087;
        double r5251089 = r5251088 / r5251080;
        return r5251089;
}
↓
double f(double alpha, double beta, double i) {
        double r5251090 = beta;
        double r5251091 = alpha;
        double r5251092 = r5251090 - r5251091;
        double r5251093 = cbrt(r5251092);
        double r5251094 = r5251091 + r5251090;
        double r5251095 = i;
        double r5251096 = 2.0;
        double r5251097 = r5251095 * r5251096;
        double r5251098 = r5251094 + r5251097;
        double r5251099 = cbrt(r5251098);
        double r5251100 = r5251093 / r5251099;
        double r5251101 = r5251100 * r5251100;
        double r5251102 = r5251096 + r5251098;
        double r5251103 = r5251102 / r5251100;
        double r5251104 = r5251101 / r5251103;
        double r5251105 = r5251104 * r5251094;
        double r5251106 = 1.0;
        double r5251107 = r5251105 + r5251106;
        double r5251108 = exp(r5251107);
        double r5251109 = log(r5251108);
        double r5251110 = r5251109 / r5251096;
        return r5251110;
}

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 23.9

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

  3. Applied *-un-lft-identity23.9

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

  4. Applied *-un-lft-identity23.9

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

  5. Applied times-frac12.1

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

  6. Applied times-frac12.0

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

  7. Simplified12.0

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

  9. Applied +-commutative12.0

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

  11. Applied add-cube-cbrt12.3

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

  12. Applied add-cube-cbrt12.1

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

  13. Applied times-frac12.1

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

  14. Applied associate-/l*12.1

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

  16. Applied add-log-exp12.1

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

  17. Applied add-log-exp12.1

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

  18. Applied sum-log12.1

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

  19. Simplified12.1

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

  20. Final simplification12.1

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

Reproduce

herbie shell --seed 2019192 
(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))