Average Error: 62.0 → 52.0
Time: 2.3s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\]
\frac{x - lo}{hi - lo}
↓
\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x) :precision binary64 (* (/ lo hi) (+ (/ x hi) -1.0)))
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) + -1.0);
}

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 58.1

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

  3. Simplified58.1

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

  4. Taylor expanded around inf 52.0

    \[\leadsto \color{blue}{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot lo}\]

  5. Simplified52.0

    \[\leadsto \color{blue}{\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity_binary6452.0

    \[\leadsto \frac{lo}{hi} \cdot \color{blue}{\left(1 \cdot \left(\frac{x}{hi} + -1\right)\right)}\]

  8. Applied associate-*r*_binary6452.0

    \[\leadsto \color{blue}{\left(\frac{lo}{hi} \cdot 1\right) \cdot \left(\frac{x}{hi} + -1\right)}\]

  9. Final simplification52.0

    \[\leadsto \frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))