Average Error: 24.2 → 12.6
Time: 11.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{{\left(\left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}^{1}}{2}\]
\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(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\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) pow(((double) (((double) (((double) (beta - alpha)) * ((double) (((double) (((double) (alpha + beta)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)))))) + 1.0)), 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 24.2

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

  3. Applied *-un-lft-identity24.2

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

  4. Applied *-un-lft-identity24.2

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

  5. Applied *-commutative24.2

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

  6. Applied times-frac12.6

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

  7. Applied times-frac12.6

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

  8. Simplified12.6

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

  10. Applied add-cbrt-cube12.6

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

  11. Simplified12.6

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

  13. Applied *-un-lft-identity12.6

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

  14. Applied pow-unpow12.6

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

  15. Applied rem-cbrt-cube12.6

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

  16. Final simplification12.6

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

Reproduce

herbie shell --seed 2020114 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))