Average Error: 62.0 → 51.9
Time: 5.8s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{hi}{lo}}\\ \left(1 + \left(\left(1 + \frac{hi}{lo}\right) \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left|t_0\right|\right) - \left(\frac{x}{lo} + \frac{hi}{lo} \cdot \frac{x}{lo}\right) \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\frac{hi}{lo}}\\
\left(1 + \left(\left(1 + \frac{hi}{lo}\right) \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left|t_0\right|\right) - \left(\frac{x}{lo} + \frac{hi}{lo} \cdot \frac{x}{lo}\right)
\end{array}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (cbrt (/ hi lo))))
   (-
    (+ 1.0 (* (* (+ 1.0 (/ hi lo)) (* t_0 t_0)) (fabs t_0)))
    (+ (/ x lo) (* (/ hi lo) (/ x lo))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	double t_0 = cbrt(hi / lo);
	return (1.0 + (((1.0 + (hi / lo)) * (t_0 * t_0)) * fabs(t_0))) - ((x / lo) + ((hi / lo) * (x / lo)));
}

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 0 64.0

    \[\leadsto \color{blue}{\left(1 + \left(\frac{hi}{lo} + \frac{{hi}^{2}}{{lo}^{2}}\right)\right) - \left(\frac{{hi}^{2} \cdot x}{{lo}^{3}} + \left(\frac{hi \cdot x}{{lo}^{2}} + \frac{x}{lo}\right)\right)} \]

  3. Simplified51.9

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

  4. Applied add-cube-cbrt_binary6451.9

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

  5. Applied associate-*r*_binary6451.9

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

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

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

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

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

  8. Applied times-frac_binary6451.9

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

  9. Applied cbrt-prod_binary6451.9

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

  10. Simplified51.9

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

  11. Simplified51.9

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

  12. Taylor expanded in hi around 0 51.9

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

  13. Final simplification51.9

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

Reproduce

herbie shell --seed 2022068 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))