Average Error: 52.4 → 0.6
Time: 9.5m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\frac{\alpha + i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{i + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
\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}
↓
\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\frac{\alpha + i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{i + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}
double f(double alpha, double beta, double i) {
        double r44303018 = i;
        double r44303019 = alpha;
        double r44303020 = beta;
        double r44303021 = r44303019 + r44303020;
        double r44303022 = r44303021 + r44303018;
        double r44303023 = r44303018 * r44303022;
        double r44303024 = r44303020 * r44303019;
        double r44303025 = r44303024 + r44303023;
        double r44303026 = r44303023 * r44303025;
        double r44303027 = 2.0;
        double r44303028 = r44303027 * r44303018;
        double r44303029 = r44303021 + r44303028;
        double r44303030 = r44303029 * r44303029;
        double r44303031 = r44303026 / r44303030;
        double r44303032 = 1.0;
        double r44303033 = r44303030 - r44303032;
        double r44303034 = r44303031 / r44303033;
        return r44303034;
}
↓
double f(double alpha, double beta, double i) {
        double r44303035 = i;
        double r44303036 = alpha;
        double r44303037 = beta;
        double r44303038 = r44303036 + r44303037;
        double r44303039 = r44303038 + r44303035;
        double r44303040 = 2.0;
        double r44303041 = r44303040 * r44303035;
        double r44303042 = r44303038 + r44303041;
        double r44303043 = r44303039 / r44303042;
        double r44303044 = r44303035 * r44303043;
        double r44303045 = 1.0;
        double r44303046 = sqrt(r44303045);
        double r44303047 = r44303046 + r44303042;
        double r44303048 = r44303044 / r44303047;
        double r44303049 = r44303036 + r44303035;
        double r44303050 = r44303042 - r44303046;
        double r44303051 = sqrt(r44303050);
        double r44303052 = r44303049 / r44303051;
        double r44303053 = r44303035 + r44303037;
        double r44303054 = r44303053 / r44303042;
        double r44303055 = r44303052 * r44303054;
        double r44303056 = r44303055 / r44303051;
        double r44303057 = r44303048 * r44303056;
        return r44303057;
}

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 52.4

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

  3. Applied add-sqr-sqrt52.4

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

  4. Applied difference-of-squares52.4

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

  5. Applied times-frac38.6

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

  6. Applied times-frac36.5

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

  8. Applied add-sqr-sqrt36.7

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

  9. Applied add-sqr-sqrt36.6

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

  10. Applied times-frac36.6

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

  12. Applied associate-*l/36.5

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

  13. Simplified36.1

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

  15. Applied *-un-lft-identity36.1

    \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\alpha + i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{i + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]

  16. Applied times-frac0.6

    \[\leadsto \frac{\color{blue}{\frac{i}{1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\alpha + i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \frac{i + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]

  17. Simplified0.6

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

  18. Final simplification0.6

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

Reproduce

herbie shell --seed 2019124 
(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)))