Average Error: 62.0 → 51.6
Time: 3.8s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\frac{x}{hi} - {\left(\frac{\sqrt[3]{lo} \cdot \sqrt[3]{lo}}{\sqrt[3]{hi} \cdot \sqrt[3]{hi}}\right)}^{3} \cdot {\left(\frac{\sqrt[3]{lo}}{\sqrt[3]{hi}}\right)}^{3} \]
\frac{x - lo}{hi - lo}
↓
\frac{x}{hi} - {\left(\frac{\sqrt[3]{lo} \cdot \sqrt[3]{lo}}{\sqrt[3]{hi} \cdot \sqrt[3]{hi}}\right)}^{3} \cdot {\left(\frac{\sqrt[3]{lo}}{\sqrt[3]{hi}}\right)}^{3}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (-
  (/ x hi)
  (*
   (pow (/ (* (cbrt lo) (cbrt lo)) (* (cbrt hi) (cbrt hi))) 3.0)
   (pow (/ (cbrt lo) (cbrt 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 (x / hi) - (pow(((cbrt(lo) * cbrt(lo)) / (cbrt(hi) * cbrt(hi))), 3.0) * pow((cbrt(lo) / cbrt(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 in hi around inf 64.0

    \[\leadsto \color{blue}{\left(\frac{x}{hi} + \left(\frac{{lo}^{2} \cdot x}{{hi}^{3}} + \frac{lo \cdot x}{{hi}^{2}}\right)\right) - \left(\frac{{lo}^{3}}{{hi}^{3}} + \left(\frac{lo}{hi} + \frac{{lo}^{2}}{{hi}^{2}}\right)\right)} \]

  3. Simplified51.9

    \[\leadsto \color{blue}{\frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \left(\frac{lo}{hi} + 1\right) \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 - \frac{lo}{hi}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right)} \]

  4. Taylor expanded in lo around 0 51.9

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \color{blue}{1} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 - \frac{lo}{hi}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]

  5. Taylor expanded in hi around 0 64.0

    \[\leadsto \frac{x}{hi} + \color{blue}{-1 \cdot \frac{{lo}^{3}}{{hi}^{3}}} \]

  6. Simplified51.6

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

  7. Applied add-cube-cbrt_binary6451.6

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

  8. Applied add-cube-cbrt_binary6451.6

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

  9. Applied times-frac_binary6451.6

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

  10. Applied unpow-prod-down_binary6451.6

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

  11. Final simplification51.6

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

Reproduce

herbie shell --seed 2021280 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))