Average Error: 24.9 → 0.9
Time: 19.2s
Precision: 64
Internal Precision: 576
\[\frac{1 - \left(b - r\right) \cdot \left(b - r\right)}{\left(4 \cdot b\right) \cdot r}\]
\[\begin{array}{l} \mathbf{if}\;r \le -86.726910540641 \lor \neg \left(r \le 1.1813564328514398 \cdot 10^{-06}\right):\\ \;\;\;\;\frac{1}{2} - \frac{1}{4} \cdot \left(\frac{r}{b} + \frac{b}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} - \frac{\frac{1}{4}}{r} \cdot \left(b - \frac{1}{b}\right)\\ \end{array}\]

Error

Bits error versus b

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if r < -86.726910540641 or 1.1813564328514398e-06 < r

    1. Initial program 35.9

      \[\frac{1 - \left(b - r\right) \cdot \left(b - r\right)}{\left(4 \cdot b\right) \cdot r}\]
    2. Initial simplification35.9

      \[\leadsto \frac{1 - \left(b - r\right) \cdot \left(b - r\right)}{b \cdot \left(r \cdot 4\right)}\]
    3. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\frac{1}{4} \cdot \frac{r}{b} + \frac{1}{4} \cdot \frac{b}{r}\right)}\]
    4. Simplified0.9

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\frac{r}{b} + \frac{b}{r}\right) \cdot \frac{1}{4}}\]

    if -86.726910540641 < r < 1.1813564328514398e-06

    1. Initial program 8.0

      \[\frac{1 - \left(b - r\right) \cdot \left(b - r\right)}{\left(4 \cdot b\right) \cdot r}\]
    2. Initial simplification8.0

      \[\leadsto \frac{1 - \left(b - r\right) \cdot \left(b - r\right)}{b \cdot \left(r \cdot 4\right)}\]
    3. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{r \cdot b} + \frac{1}{2}\right) - \frac{1}{4} \cdot \frac{b}{r}}\]
    4. Simplified0.9

      \[\leadsto \color{blue}{\frac{1}{2} - \left(b - \frac{1}{b}\right) \cdot \frac{\frac{1}{4}}{r}}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;r \le -86.726910540641 \lor \neg \left(r \le 1.1813564328514398 \cdot 10^{-06}\right):\\ \;\;\;\;\frac{1}{2} - \frac{1}{4} \cdot \left(\frac{r}{b} + \frac{b}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} - \frac{\frac{1}{4}}{r} \cdot \left(b - \frac{1}{b}\right)\\ \end{array}\]

Runtime

Time bar (total: 19.2s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (b r)
  :name "(1-(b-r)*(b-r))/(4*b*r)"
  (/ (- 1 (* (- b r) (- b r))) (* (* 4 b) r)))