Average Error: 62.0 → 51.5
Time: 4.9s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\frac{hi}{lo} \cdot \frac{hi}{lo} - hi \cdot \frac{x}{lo \cdot lo} \]
\frac{x - lo}{hi - lo}
↓
\frac{hi}{lo} \cdot \frac{hi}{lo} - hi \cdot \frac{x}{lo \cdot lo}

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

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 in lo around inf 64.0

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

  3. Simplified51.9

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

  4. Taylor expanded in hi around inf 57.9

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

  5. Simplified51.9

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

  6. Taylor expanded in hi around inf 64.0

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} - \frac{hi}{lo} \cdot \frac{x}{lo} \]

  7. Simplified51.5

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

  8. Applied div-inv_binary6451.5

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

  9. Applied associate-*l*_binary6451.5

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

  10. Simplified51.5

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

  11. Final simplification51.5

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

Reproduce

herbie shell --seed 2021224 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))