Average Error: 62.0 → 50.8
Time: 6.0s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\sqrt[3]{\frac{x}{hi} + \log \left(1 - \frac{lo}{hi}\right)} \cdot \left(\sqrt[3]{\frac{x}{hi} + \log \left(1 - \frac{lo}{hi}\right)} \cdot \sqrt[3]{\frac{x}{hi} + \log \left(1 - \frac{lo}{hi}\right)}\right)\]
\frac{x - lo}{hi - lo}
↓
\sqrt[3]{\frac{x}{hi} + \log \left(1 - \frac{lo}{hi}\right)} \cdot \left(\sqrt[3]{\frac{x}{hi} + \log \left(1 - \frac{lo}{hi}\right)} \cdot \sqrt[3]{\frac{x}{hi} + \log \left(1 - \frac{lo}{hi}\right)}\right)
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (*
  (cbrt (+ (/ x hi) (log (- 1.0 (/ lo hi)))))
  (*
   (cbrt (+ (/ x hi) (log (- 1.0 (/ lo hi)))))
   (cbrt (+ (/ x hi) (log (- 1.0 (/ lo hi))))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return cbrt((x / hi) + log(1.0 - (lo / hi))) * (cbrt((x / hi) + log(1.0 - (lo / hi))) * cbrt((x / hi) + log(1.0 - (lo / 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-log-exp_binary64_79952.0

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

  5. Taylor expanded around 0 50.8

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

  6. Simplified50.8

    \[\leadsto \log \color{blue}{\left(\left(1 - \frac{lo}{hi}\right) \cdot e^{\frac{x}{hi}}\right)}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt_binary64_79550.8

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

  9. Simplified50.8

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

  10. Simplified50.8

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

  11. Final simplification50.8

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

Reproduce

herbie shell --seed 2021093 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))