Average Error: 62.0 → 51.5
Time: 4.0s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\left(1 + \frac{hi}{lo} \cdot \sqrt{\log \left(e^{{\left(1 + \frac{hi}{lo}\right)}^{2}}\right)}\right) - \frac{x}{lo} \]
\frac{x - lo}{hi - lo}
↓
\left(1 + \frac{hi}{lo} \cdot \sqrt{\log \left(e^{{\left(1 + \frac{hi}{lo}\right)}^{2}}\right)}\right) - \frac{x}{lo}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (-
  (+ 1.0 (* (/ hi lo) (sqrt (log (exp (pow (+ 1.0 (/ hi lo)) 2.0))))))
  (/ x lo)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (1.0 + ((hi / lo) * sqrt(log(exp(pow((1.0 + (hi / lo)), 2.0)))))) - (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 egg51.5

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

  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 egg51.5

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

  7. Final simplification51.5

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

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)))