Average Error: 62.0 → 52.0
Time: 7.1s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\sqrt[3]{\sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}} \cdot \left(\sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}}\right)}\]
\frac{x - lo}{hi - lo}
↓
\sqrt[3]{\sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}} \cdot \left(\sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}}\right)}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (cbrt
  (*
   (cbrt (pow (/ (- x lo) hi) 3.0))
   (* (cbrt (pow (/ (- x lo) hi) 3.0)) (cbrt (pow (/ (- x lo) hi) 3.0))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return cbrt(cbrt(pow(((x - lo) / hi), 3.0)) * (cbrt(pow(((x - lo) / hi), 3.0)) * cbrt(pow(((x - lo) / hi), 3.0))));
}

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-cbrt-cube_binary64_79652.0

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

  5. Simplified52.0

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

  7. Applied add-cube-cbrt_binary64_79552.0

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

  8. Final simplification52.0

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

Reproduce

herbie shell --seed 2021045 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))