Average Error: 62.0 → 51.9
Time: 3.1s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \frac{-1}{{hi}^{0.6666666666666666}} \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]
\frac{x - lo}{hi - lo}
↓
\frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \frac{-1}{{hi}^{0.6666666666666666}} \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right)

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (+
  (/ x hi)
  (-
   (fma
    (/ x hi)
    (* (/ lo hi) (/ lo hi))
    (*
     (/ lo hi)
     (+ -1.0 (* (/ -1.0 (pow hi 0.6666666666666666)) (/ lo (cbrt hi))))))
   (pow (/ 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 (x / hi) + (fma((x / hi), ((lo / hi) * (lo / hi)), ((lo / hi) * (-1.0 + ((-1.0 / pow(hi, 0.6666666666666666)) * (lo / cbrt(hi)))))) - pow((lo / hi), 3.0));
}

Error

Bits error versus lo

Bits error versus hi

Bits error versus x

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 inf 51.9

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

  5. Applied add-cube-cbrt_binary6451.9

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

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

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

  7. Applied times-frac_binary6451.9

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

  8. Applied cancel-sign-sub-inv_binary6451.9

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

  9. Applied pow1/3_binary6451.9

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \left(-\frac{1}{\sqrt[3]{hi} \cdot \color{blue}{{hi}^{0.3333333333333333}}}\right) \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]

  10. Applied pow1/3_binary6451.9

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \left(-\frac{1}{\color{blue}{{hi}^{0.3333333333333333}} \cdot {hi}^{0.3333333333333333}}\right) \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]

  11. Applied pow-prod-up_binary6451.9

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \left(-\frac{1}{\color{blue}{{hi}^{\left(0.3333333333333333 + 0.3333333333333333\right)}}}\right) \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]

  12. Simplified51.9

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \left(-\frac{1}{{hi}^{\color{blue}{0.6666666666666666}}}\right) \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]

  13. Final simplification51.9

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 + \frac{-1}{{hi}^{0.6666666666666666}} \cdot \frac{lo}{\sqrt[3]{hi}}\right)\right) - {\left(\frac{lo}{hi}\right)}^{3}\right) \]

Reproduce

herbie shell --seed 2022076 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))