Average Error: 62.0 → 52.0
Time: 2.7s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\frac{lo}{\frac{hi}{\frac{x}{hi}}} + e^{\log \left(x - lo\right) - \log hi}\]
\frac{x - lo}{hi - lo}
↓
\frac{lo}{\frac{hi}{\frac{x}{hi}}} + e^{\log \left(x - lo\right) - \log hi}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (+ (/ lo (/ hi (/ x hi))) (exp (- (log (- x lo)) (log hi)))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (lo / (hi / (x / hi))) + exp(log(x - lo) - log(hi));
}

Error

Bits error versus lo

Bits error versus hi

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 62.0

    \[\frac{x - lo}{hi - lo}\]

  2. Taylor expanded around 0 58.0

    \[\leadsto \color{blue}{\left(\frac{x \cdot lo}{{hi}^{2}} + \frac{x}{hi}\right) - \frac{lo}{hi}}\]

  3. Simplified58.0

    \[\leadsto \color{blue}{\frac{lo \cdot x}{hi \cdot hi} + \frac{x - lo}{hi}}\]
  4. Using strategy rm

  5. Applied associate-/l*_binary6452.0

    \[\leadsto \color{blue}{\frac{lo}{\frac{hi \cdot hi}{x}}} + \frac{x - lo}{hi}\]

  6. Simplified52.0

    \[\leadsto \frac{lo}{\color{blue}{\frac{hi}{\frac{x}{hi}}}} + \frac{x - lo}{hi}\]
  7. Using strategy rm

  8. Applied add-exp-log_binary6452.0

    \[\leadsto \frac{lo}{\frac{hi}{\frac{x}{hi}}} + \frac{x - lo}{\color{blue}{e^{\log hi}}}\]

  9. Applied add-exp-log_binary6452.0

    \[\leadsto \frac{lo}{\frac{hi}{\frac{x}{hi}}} + \frac{\color{blue}{e^{\log \left(x - lo\right)}}}{e^{\log hi}}\]

  10. Applied div-exp_binary6452.0

    \[\leadsto \frac{lo}{\frac{hi}{\frac{x}{hi}}} + \color{blue}{e^{\log \left(x - lo\right) - \log hi}}\]

  11. Final simplification52.0

    \[\leadsto \frac{lo}{\frac{hi}{\frac{x}{hi}}} + e^{\log \left(x - lo\right) - \log hi}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))