Average Error: 24.3 → 12.5
Time: 8.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\left(\alpha + \beta\right) \cdot \left(\frac{1}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right) + 1}{2}\]
\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(\alpha + \beta\right) \cdot \left(\frac{1}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}\right) + 1}{2}
double code(double alpha, double beta, double i) {
	return (((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) (alpha + beta)) * ((double) ((1.0 / ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))) * (((double) (beta - alpha)) / ((double) (beta + ((double) (alpha + ((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 24.3

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

  2. Simplified20.0

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

  4. Applied *-un-lft-identity20.0

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

  5. Applied times-frac12.5

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

  6. Simplified12.5

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

  7. Simplified12.5

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

  8. Final simplification12.5

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

Reproduce

herbie shell --seed 2020199 
(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))