Average Error: 62.0 → 51.9
Time: 3.2s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\left(1 + \mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right) \cdot \frac{1}{lo}\right) - \left(\frac{x}{lo} + \frac{hi}{lo} \cdot \left(\frac{hi}{lo} \cdot \frac{x}{lo}\right)\right) \]
\frac{x - lo}{hi - lo}
↓
\left(1 + \mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right) \cdot \frac{1}{lo}\right) - \left(\frac{x}{lo} + \frac{hi}{lo} \cdot \left(\frac{hi}{lo} \cdot \frac{x}{lo}\right)\right)

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

Error

Bits error versus lo

Bits error versus hi

Bits error versus x

Derivation

  1. Initial program 62.0

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

  2. Taylor expanded in hi around 0 64.0

    \[\leadsto \color{blue}{\left(1 + \left(\frac{hi}{lo} + \frac{{hi}^{2}}{{lo}^{2}}\right)\right) - \left(\frac{{hi}^{2} \cdot x}{{lo}^{3}} + \left(\frac{hi \cdot x}{{lo}^{2}} + \frac{x}{lo}\right)\right)} \]

  3. Simplified51.9

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

  4. Taylor expanded in hi around inf 51.9

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

  5. Applied div-inv_binary6451.9

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

  6. Applied associate-*r*_binary6451.9

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

  7. Simplified51.9

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

  8. Applied associate-*r*_binary6451.9

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

  9. Simplified51.9

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

  10. Final simplification51.9

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

Reproduce

herbie shell --seed 2022077 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))