Average Error: 62.0 → 52.0
Time: 2.0s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[lo \cdot \frac{\frac{x}{hi} - 1}{hi} \]
\frac{x - lo}{hi - lo}
↓
lo \cdot \frac{\frac{x}{hi} - 1}{hi}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x) :precision binary64 (* lo (/ (- (/ x hi) 1.0) hi)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return lo * (((x / hi) - 1.0) / 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

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

  3. Simplified58.0

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

  4. Taylor expanded around inf 52.0

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

  5. Simplified52.0

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

  7. Applied div-inv_binary6452.0

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

  8. Applied associate-*l*_binary6452.0

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

  9. Simplified52.0

    \[\leadsto lo \cdot \color{blue}{\frac{\frac{x}{hi} - 1}{hi}} \]

  10. Final simplification52.0

    \[\leadsto lo \cdot \frac{\frac{x}{hi} - 1}{hi} \]

Reproduce

herbie shell --seed 2021205 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))