Average Error: 52.5 → 36.4
Time: 48.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\left(\frac{\sqrt{\frac{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}
↓
\left(\frac{\sqrt{\frac{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)
double f(double alpha, double beta, double i) {
        double r5537801 = i;
        double r5537802 = alpha;
        double r5537803 = beta;
        double r5537804 = r5537802 + r5537803;
        double r5537805 = r5537804 + r5537801;
        double r5537806 = r5537801 * r5537805;
        double r5537807 = r5537803 * r5537802;
        double r5537808 = r5537807 + r5537806;
        double r5537809 = r5537806 * r5537808;
        double r5537810 = 2.0;
        double r5537811 = r5537810 * r5537801;
        double r5537812 = r5537804 + r5537811;
        double r5537813 = r5537812 * r5537812;
        double r5537814 = r5537809 / r5537813;
        double r5537815 = 1.0;
        double r5537816 = r5537813 - r5537815;
        double r5537817 = r5537814 / r5537816;
        return r5537817;
}
↓
double f(double alpha, double beta, double i) {
        double r5537818 = beta;
        double r5537819 = alpha;
        double r5537820 = r5537818 * r5537819;
        double r5537821 = r5537819 + r5537818;
        double r5537822 = i;
        double r5537823 = r5537821 + r5537822;
        double r5537824 = r5537823 * r5537822;
        double r5537825 = r5537820 + r5537824;
        double r5537826 = 2.0;
        double r5537827 = r5537822 * r5537826;
        double r5537828 = r5537821 + r5537827;
        double r5537829 = r5537825 / r5537828;
        double r5537830 = sqrt(r5537829);
        double r5537831 = 1.0;
        double r5537832 = sqrt(r5537831);
        double r5537833 = r5537828 - r5537832;
        double r5537834 = sqrt(r5537833);
        double r5537835 = r5537830 / r5537834;
        double r5537836 = r5537824 / r5537828;
        double r5537837 = sqrt(r5537836);
        double r5537838 = r5537832 + r5537828;
        double r5537839 = sqrt(r5537838);
        double r5537840 = r5537837 / r5537839;
        double r5537841 = r5537835 * r5537840;
        double r5537842 = r5537841 * r5537841;
        return r5537842;
}

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 52.5

    \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt52.5

    \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

  4. Applied difference-of-squares52.5

    \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

  5. Applied times-frac38.2

    \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

  6. Applied times-frac36.3

    \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt36.5

    \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}}\]

  9. Applied add-sqr-sqrt36.4

    \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]

  10. Applied times-frac36.4

    \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \color{blue}{\left(\frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)}\]

  11. Applied add-sqr-sqrt36.5

    \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}} \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)\]

  12. Applied add-sqr-sqrt36.4

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}} \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)\]

  13. Applied times-frac36.4

    \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}\right)} \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)\]

  14. Applied unswap-sqr36.4

    \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right) \cdot \left(\frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)}\]

  15. Final simplification36.4

    \[\leadsto \left(\frac{\sqrt{\frac{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \left(\frac{\sqrt{\frac{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\]

Reproduce

herbie shell --seed 2019151 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))