Average Error: 23.9 → 11.8
Time: 18.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{1}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]
\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{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{1}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}
double code(double alpha, double beta, double i) {
	return ((((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta, double i) {
	return ((((beta / ((fma(i, 2.0, (alpha + beta)) / (cbrt((alpha + beta)) * cbrt((alpha + beta)))) * (1.0 / cbrt((alpha + beta))))) / (((alpha + beta) + (2.0 * i)) + 2.0)) - (((alpha / (fma(i, 2.0, (alpha + beta)) / (alpha + beta))) / (((alpha + beta) + (2.0 * i)) + 2.0)) - 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 23.9

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

  3. Applied *-un-lft-identity23.9

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

  4. Applied associate-/r*23.9

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

  5. Simplified12.0

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

  7. Applied div-sub12.0

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

  8. Applied div-sub12.0

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

  9. Applied associate-+l-11.7

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

  11. Applied div-inv11.7

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

  13. Applied add-cube-cbrt11.8

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

  14. Applied *-un-lft-identity11.8

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

  15. Applied times-frac11.8

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

  16. Applied associate-*r*11.8

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

  17. Simplified11.8

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

  18. Final simplification11.8

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))