Average Error: 62.0 → 52.0
Time: 4.2s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\frac{\sqrt{x - lo}}{\sqrt[3]{hi} \cdot \sqrt[3]{hi}} \cdot \frac{\sqrt{x - lo}}{{\left(e^{\sqrt[3]{\log \left(\sqrt[3]{hi}\right)} \cdot \sqrt[3]{\log \left(\sqrt[3]{hi}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\sqrt[3]{hi}\right)}\right)}}\]
\frac{x - lo}{hi - lo}
↓
\frac{\sqrt{x - lo}}{\sqrt[3]{hi} \cdot \sqrt[3]{hi}} \cdot \frac{\sqrt{x - lo}}{{\left(e^{\sqrt[3]{\log \left(\sqrt[3]{hi}\right)} \cdot \sqrt[3]{\log \left(\sqrt[3]{hi}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\sqrt[3]{hi}\right)}\right)}}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (*
  (/ (sqrt (- x lo)) (* (cbrt hi) (cbrt hi)))
  (/
   (sqrt (- x lo))
   (pow
    (exp (* (cbrt (log (cbrt hi))) (cbrt (log (cbrt hi)))))
    (cbrt (log (cbrt hi)))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (sqrt(x - lo) / (cbrt(hi) * cbrt(hi))) * (sqrt(x - lo) / pow(exp(cbrt(log(cbrt(hi))) * cbrt(log(cbrt(hi)))), cbrt(log(cbrt(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-cube-cbrt_binary64_79552.0

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

  5. Applied add-sqr-sqrt_binary64_78252.0

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

  6. Applied times-frac_binary64_76652.0

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

  8. Applied add-exp-log_binary64_79852.0

    \[\leadsto \frac{\sqrt{x - lo}}{\sqrt[3]{hi} \cdot \sqrt[3]{hi}} \cdot \frac{\sqrt{x - lo}}{\color{blue}{e^{\log \left(\sqrt[3]{hi}\right)}}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt_binary64_79552.0

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

  11. Applied exp-prod_binary64_81252.0

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

  12. Final simplification52.0

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

Reproduce

herbie shell --seed 2021015 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))