Average Error: 62.0 → 50.4
Time: 6.1s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{e^{\frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}}}\\ \left(1 + \left(\log \left(t_0 \cdot t_0\right) + \frac{hi}{lo} \cdot 0.3333333333333333\right)\right) - \left(\frac{x}{lo} + \frac{x}{lo} \cdot \left(\frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)\right) \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{e^{\frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}}}\\
\left(1 + \left(\log \left(t_0 \cdot t_0\right) + \frac{hi}{lo} \cdot 0.3333333333333333\right)\right) - \left(\frac{x}{lo} + \frac{x}{lo} \cdot \left(\frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)\right)
\end{array}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (/ (fma hi (/ hi lo) hi) lo)))))
   (-
    (+ 1.0 (+ (log (* t_0 t_0)) (* (/ hi lo) 0.3333333333333333)))
    (+ (/ x lo) (* (/ x lo) (* (/ hi lo) (+ 1.0 (/ hi 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(exp(fma(hi, (hi / lo), hi) / lo));
	return (1.0 + (log(t_0 * t_0) + ((hi / lo) * 0.3333333333333333))) - ((x / lo) + ((x / lo) * ((hi / lo) * (1.0 + (hi / lo)))));
}

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 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-log-exp_binary6451.9

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

  5. Simplified51.9

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

  6. Applied add-cube-cbrt_binary6451.9

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

  7. Applied log-prod_binary6451.9

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

  8. Taylor expanded in hi around 0 50.4

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

  9. Final simplification50.4

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

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)))