Average Error: 24.0 → 12.5
Time: 6.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\left(\left(\alpha + \beta\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2}\]
\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 \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2}
(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))
↓
(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (*
    (*
     (+ alpha beta)
     (/
      (* (cbrt (- beta alpha)) (cbrt (- beta alpha)))
      (*
       (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
       (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))))
    (/
     (/ (cbrt (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
     (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
   1.0)
  2.0))
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 ((((alpha + beta) * ((cbrt(beta - alpha) * cbrt(beta - alpha)) / (cbrt(2.0 + ((alpha + beta) + (2.0 * i))) * cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))) * ((cbrt(beta - alpha) / ((alpha + beta) + (2.0 * i))) / cbrt(2.0 + ((alpha + beta) + (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 24.0

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

  3. Applied *-un-lft-identity_binary6424.0

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

  4. Applied *-un-lft-identity_binary6424.0

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

  5. Applied times-frac_binary6412.4

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

  6. Applied times-frac_binary6412.4

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

  7. Simplified12.4

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

  8. Simplified12.4

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

  10. Applied add-cube-cbrt_binary6412.7

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

  11. Applied *-un-lft-identity_binary6412.7

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

  12. Applied add-cube-cbrt_binary6412.5

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

  13. Applied times-frac_binary6412.5

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

  14. Applied times-frac_binary6412.5

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

  15. Applied associate-*r*_binary6412.5

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

  16. Simplified12.5

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

  17. Final simplification12.5

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

Reproduce

herbie shell --seed 2020219 
(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))