Average Error: 62.0 → 51.2
Time: 2.8s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\mathsf{expm1}\left(\log 2 + \frac{hi}{lo} \cdot \left(0.5 + \frac{hi \cdot 0.375}{lo}\right)\right) - \left(\frac{hi}{lo} + 1\right) \cdot \frac{x}{lo} \]
\frac{x - lo}{hi - lo}
↓
\mathsf{expm1}\left(\log 2 + \frac{hi}{lo} \cdot \left(0.5 + \frac{hi \cdot 0.375}{lo}\right)\right) - \left(\frac{hi}{lo} + 1\right) \cdot \frac{x}{lo}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (-
  (expm1 (+ (log 2.0) (* (/ hi lo) (+ 0.5 (/ (* hi 0.375) lo)))))
  (* (+ (/ hi lo) 1.0) (/ x lo))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return expm1(log(2.0) + ((hi / lo) * (0.5 + ((hi * 0.375) / lo)))) - (((hi / lo) + 1.0) * (x / 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. Applied expm1-log1p-u_binary6451.9

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(1 + \frac{hi}{lo}\right) \cdot \frac{hi}{lo}\right)\right)} - \left(1 + \frac{hi}{lo}\right) \cdot \frac{x}{lo} \]

  5. Simplified51.9

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{hi}{lo}, 1 + \frac{hi}{lo}, 1\right)\right)}\right) - \left(1 + \frac{hi}{lo}\right) \cdot \frac{x}{lo} \]

  6. Taylor expanded in hi around 0 64.0

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 + \left(0.5 \cdot \frac{hi}{lo} + 0.375 \cdot \frac{{hi}^{2}}{{lo}^{2}}\right)}\right) - \left(1 + \frac{hi}{lo}\right) \cdot \frac{x}{lo} \]

  7. Simplified51.2

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 + \frac{hi}{lo} \cdot \left(0.5 + \frac{hi \cdot 0.375}{lo}\right)}\right) - \left(1 + \frac{hi}{lo}\right) \cdot \frac{x}{lo} \]

  8. Final simplification51.2

    \[\leadsto \mathsf{expm1}\left(\log 2 + \frac{hi}{lo} \cdot \left(0.5 + \frac{hi \cdot 0.375}{lo}\right)\right) - \left(\frac{hi}{lo} + 1\right) \cdot \frac{x}{lo} \]

Reproduce

herbie shell --seed 2022061 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))