Average Error: 62.0 → 51.9
Time: 3.6s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\left(1 + \frac{hi}{lo} \cdot \log \left(e^{1 + \frac{hi}{lo}}\right)\right) - \left(\frac{x}{lo} + \frac{x}{lo} \cdot \left(\left(hi \cdot \frac{\sqrt[3]{hi} \cdot \sqrt[3]{hi}}{lo}\right) \cdot \frac{\sqrt[3]{hi}}{lo}\right)\right) \]
\frac{x - lo}{hi - lo}
↓
\left(1 + \frac{hi}{lo} \cdot \log \left(e^{1 + \frac{hi}{lo}}\right)\right) - \left(\frac{x}{lo} + \frac{x}{lo} \cdot \left(\left(hi \cdot \frac{\sqrt[3]{hi} \cdot \sqrt[3]{hi}}{lo}\right) \cdot \frac{\sqrt[3]{hi}}{lo}\right)\right)

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (-
  (+ 1.0 (* (/ hi lo) (log (exp (+ 1.0 (/ hi lo))))))
  (+
   (/ x lo)
   (* (/ x lo) (* (* hi (/ (* (cbrt hi) (cbrt hi)) lo)) (/ (cbrt hi) 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) * log(exp(1.0 + (hi / lo))))) - ((x / lo) + ((x / lo) * ((hi * ((cbrt(hi) * cbrt(hi)) / lo)) * (cbrt(hi) / 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 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 add-log-exp_binary6451.9

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

  6. Applied add-log-exp_binary6451.9

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

  7. Applied sum-log_binary6451.9

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

  8. Simplified51.9

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

  9. Applied *-un-lft-identity_binary6451.9

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

  10. Applied add-cube-cbrt_binary6451.9

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

  11. Applied times-frac_binary6451.9

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

  12. Applied associate-*r*_binary6451.9

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

  13. Simplified51.9

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

  14. Final simplification51.9

    \[\leadsto \left(1 + \frac{hi}{lo} \cdot \log \left(e^{1 + \frac{hi}{lo}}\right)\right) - \left(\frac{x}{lo} + \frac{x}{lo} \cdot \left(\left(hi \cdot \frac{\sqrt[3]{hi} \cdot \sqrt[3]{hi}}{lo}\right) \cdot \frac{\sqrt[3]{hi}}{lo}\right)\right) \]

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)))