Average Error: 62.0 → 51.9
Time: 6.5s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log hi}\\ \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right), \frac{lo}{hi} \cdot \left(-1 - \frac{lo}{hi}\right)\right) - {\left({\left(e^{\mathsf{fma}\left(t_0 \cdot t_0, -t_0, \log \left(-lo\right)\right)}\right)}^{0.6666666666666666}\right)}^{3} \cdot {\left(\sqrt[3]{\frac{lo}{hi}}\right)}^{3}\right) \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\log hi}\\
\frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right), \frac{lo}{hi} \cdot \left(-1 - \frac{lo}{hi}\right)\right) - {\left({\left(e^{\mathsf{fma}\left(t_0 \cdot t_0, -t_0, \log \left(-lo\right)\right)}\right)}^{0.6666666666666666}\right)}^{3} \cdot {\left(\sqrt[3]{\frac{lo}{hi}}\right)}^{3}\right)
\end{array}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (cbrt (log hi))))
   (+
    (/ x hi)
    (-
     (fma
      (/ x hi)
      (* (/ lo hi) (+ (/ lo hi) 1.0))
      (* (/ lo hi) (- -1.0 (/ lo hi))))
     (*
      (pow
       (pow (exp (fma (* t_0 t_0) (- t_0) (log (- lo)))) 0.6666666666666666)
       3.0)
      (pow (cbrt (/ 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) {
	double t_0 = cbrt(log(hi));
	return (x / hi) + (fma((x / hi), ((lo / hi) * ((lo / hi) + 1.0)), ((lo / hi) * (-1.0 - (lo / hi)))) - (pow(pow(exp(fma((t_0 * t_0), -t_0, log(-lo))), 0.6666666666666666), 3.0) * pow(cbrt(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. Applied add-cube-cbrt_binary6451.9

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

  5. Applied unpow-prod-down_binary6451.9

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

  6. Taylor expanded in lo around -inf 58.0

    \[\leadsto \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \left(\frac{lo}{hi} + 1\right) \cdot \frac{lo}{hi}, \frac{lo}{hi} \cdot \left(-1 - \frac{lo}{hi}\right)\right) - \color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot {\left(e^{0.3333333333333333 \cdot \left(\log \left(\frac{1}{{hi}^{2}}\right) - 2 \cdot \log \left(\frac{-1}{lo}\right)\right)}\right)}^{3}\right)} \cdot {\left(\sqrt[3]{\frac{lo}{hi}}\right)}^{3}\right) \]

  7. Simplified51.9

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

  8. Applied add-cube-cbrt_binary6451.9

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

  9. Applied distribute-rgt-neg-in_binary6451.9

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

  10. Applied fma-neg_binary6451.9

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

  11. Simplified51.9

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

  12. Final simplification51.9

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

Reproduce

herbie shell --seed 2022081 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))