Average Error: 62.0 → 51.5
Time: 3.3s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := 1 + \frac{hi}{lo}\\ \left(1 + \frac{hi}{lo} \cdot \sqrt{\mathsf{fma}\left(hi, \frac{1}{lo} \cdot t_0, t_0\right)}\right) - \frac{x}{lo} \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := 1 + \frac{hi}{lo}\\
\left(1 + \frac{hi}{lo} \cdot \sqrt{\mathsf{fma}\left(hi, \frac{1}{lo} \cdot t_0, t_0\right)}\right) - \frac{x}{lo}
\end{array}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ hi lo))))
   (- (+ 1.0 (* (/ hi lo) (sqrt (fma hi (* (/ 1.0 lo) t_0) t_0)))) (/ x lo))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	double t_0 = 1.0 + (hi / lo);
	return (1.0 + ((hi / lo) * sqrt(fma(hi, ((1.0 / lo) * t_0), t_0)))) - (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 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. Applied egg-rr51.5

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

  5. Taylor expanded in hi around 0 51.5

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

  6. Applied egg-rr51.5

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

  7. Final simplification51.5

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

Reproduce

herbie shell --seed 2022125 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))