Average Error: 62.0 → 52.0
Time: 4.6s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\frac{1}{\sqrt{hi}} \cdot e^{\log \left(\frac{x - lo}{\sqrt{hi}}\right)}\]
\frac{x - lo}{hi - lo}
↓
\frac{1}{\sqrt{hi}} \cdot e^{\log \left(\frac{x - lo}{\sqrt{hi}}\right)}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (* (/ 1.0 (sqrt hi)) (exp (log (/ (- x lo) (sqrt hi))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (1.0 / sqrt(hi)) * exp(log((x - lo) / sqrt(hi)));
}

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 around inf 52.0

    \[\leadsto \color{blue}{\frac{x - lo}{hi}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary64_78252.0

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

  5. Applied *-un-lft-identity_binary64_76052.0

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

  6. Applied times-frac_binary64_76652.0

    \[\leadsto \color{blue}{\frac{1}{\sqrt{hi}} \cdot \frac{x - lo}{\sqrt{hi}}}\]
  7. Using strategy rm

  8. Applied add-exp-log_binary64_79852.0

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

  9. Applied add-exp-log_binary64_79852.0

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

  10. Applied div-exp_binary64_81152.0

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

  11. Simplified52.0

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

  12. Final simplification52.0

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

Reproduce

herbie shell --seed 2020353 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))