Average Error: 54.3 → 13.4
Time: 11.8s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \left(0.5 + \left(0.125 \cdot \frac{1}{i \cdot i} - 0.25 \cdot \frac{\sqrt{1}}{i}\right)\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i \cdot 0.5 + \left(0.25 \cdot \sqrt{1} + 0.125 \cdot \frac{1}{i}\right)}}\]
\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}
↓
\left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \left(0.5 + \left(0.125 \cdot \frac{1}{i \cdot i} - 0.25 \cdot \frac{\sqrt{1}}{i}\right)\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i \cdot 0.5 + \left(0.25 \cdot \sqrt{1} + 0.125 \cdot \frac{1}{i}\right)}}
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (i * ((double) (((double) (alpha + beta)) + i)))) * ((double) (((double) (beta * alpha)) + ((double) (i * ((double) (((double) (alpha + beta)) + i)))))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))))) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) - 1.0))));
}
↓
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (i / ((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))))) * ((double) (0.5 + ((double) (((double) (0.125 * ((double) (1.0 / ((double) (i * i)))))) - ((double) (0.25 * ((double) (((double) sqrt(1.0)) / i)))))))))) * ((double) (((double) cbrt(1.0)) / ((double) (((double) (alpha + ((double) (beta + ((double) (i * 2.0)))))) / ((double) (((double) (i * 0.5)) + ((double) (((double) (0.25 * ((double) sqrt(1.0)))) + ((double) (0.125 * ((double) (1.0 / i))))))))))))));
}

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 54.3

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

  2. Simplified49.0

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

  4. Applied clear-num49.0

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

  5. Simplified39.9

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

  7. Applied add-sqr-sqrt39.9

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

  8. Applied unpow239.9

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

  9. Applied difference-of-squares39.9

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

  10. Applied *-un-lft-identity39.9

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

  11. Applied times-frac38.6

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

  12. Applied *-un-lft-identity38.6

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

  13. Applied times-frac38.6

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

  14. Applied add-cube-cbrt38.6

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

  15. Applied times-frac38.4

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

  16. Applied associate-*r*37.9

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

  17. Simplified37.9

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

  18. Taylor expanded around inf 13.4

    \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\alpha + \left(\left(\beta + i \cdot 2\right) + \sqrt{1}\right)}\right)\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{\color{blue}{0.5 \cdot i + \left(0.25 \cdot \sqrt{1} + 0.125 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{i}\right)}}}\]

  19. Simplified13.4

    \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\alpha + \left(\left(\beta + i \cdot 2\right) + \sqrt{1}\right)}\right)\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{\color{blue}{i \cdot 0.5 + \left(\sqrt{1} \cdot 0.25 + 0.125 \cdot \frac{1}{i}\right)}}}\]

  20. Taylor expanded around inf 13.4

    \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{{i}^{2}} + 0.5\right) - 0.25 \cdot \frac{\sqrt{1}}{i}\right)}\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i \cdot 0.5 + \left(\sqrt{1} \cdot 0.25 + 0.125 \cdot \frac{1}{i}\right)}}\]

  21. Simplified13.4

    \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \color{blue}{\left(0.5 + \left(0.125 \cdot \frac{1}{i \cdot i} - 0.25 \cdot \frac{\sqrt{1}}{i}\right)\right)}\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i \cdot 0.5 + \left(\sqrt{1} \cdot 0.25 + 0.125 \cdot \frac{1}{i}\right)}}\]

  22. Final simplification13.4

    \[\leadsto \left(\frac{i}{\alpha + \left(\beta + i \cdot 2\right)} \cdot \left(0.5 + \left(0.125 \cdot \frac{1}{i \cdot i} - 0.25 \cdot \frac{\sqrt{1}}{i}\right)\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{i \cdot 0.5 + \left(0.25 \cdot \sqrt{1} + 0.125 \cdot \frac{1}{i}\right)}}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))