Average Error: 23.8 → 12.7
Time: 9.8s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\left(\left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right)\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\alpha + 2 \cdot i\right)}}{\sqrt[3]{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} + 1}{2}\]
\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{\left(\left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}\right)\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\alpha + 2 \cdot i\right)}}{\sqrt[3]{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}} + 1}{2}
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (alpha + beta)) * ((double) (beta - alpha)))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)))) + 1.0)) / 2.0));
}
↓
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (((double) (alpha + beta)) * ((double) (((double) (((double) cbrt(((double) (beta - alpha)))) / ((double) cbrt(((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i)))))))))))) * ((double) (((double) cbrt(((double) (beta - alpha)))) / ((double) cbrt(((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i)))))))))))))))) * ((double) (((double) (((double) cbrt(((double) (beta - alpha)))) / ((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))))) / ((double) cbrt(((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i)))))))))))))) + 1.0)) / 2.0));
}

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 23.8

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

  2. Simplified19.5

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

  4. Applied associate-/r*12.7

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

  5. Simplified12.7

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

  7. Applied add-cube-cbrt12.9

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

  8. Applied *-un-lft-identity12.9

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

  9. Applied add-cube-cbrt12.7

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

  10. Applied times-frac12.7

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

  11. Applied times-frac12.7

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

  12. Applied associate-*r*12.7

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

  13. Simplified12.7

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

  14. Final simplification12.7

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

Reproduce

herbie shell --seed 2020185 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :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))