Average Error: 62.0 → 50.4
Time: 4.6s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := 1 + \frac{hi}{lo}\\ t_1 := \sqrt[3]{e^{1 + \frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}}}\\ \log \left(t_1 \cdot \left(t_1 \cdot \sqrt[3]{e^{t_0}}\right)\right) - t_0 \cdot \frac{x}{lo} \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := 1 + \frac{hi}{lo}\\
t_1 := \sqrt[3]{e^{1 + \frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}}}\\
\log \left(t_1 \cdot \left(t_1 \cdot \sqrt[3]{e^{t_0}}\right)\right) - t_0 \cdot \frac{x}{lo}
\end{array}

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ hi lo)))
        (t_1 (cbrt (exp (+ 1.0 (/ (fma hi (/ hi lo) hi) lo))))))
   (- (log (* t_1 (* t_1 (cbrt (exp t_0))))) (* t_0 (/ 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 = 1.0 + (hi / lo);
	double t_1 = cbrt(exp((1.0 + (fma(hi, (hi / lo), hi) / lo))));
	return log((t_1 * (t_1 * cbrt(exp(t_0))))) - (t_0 * (x / 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 lo around inf 64.0

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

  3. Simplified51.9

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

  4. Applied add-log-exp_binary6451.9

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

  5. Applied add-log-exp_binary6451.9

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

  6. Applied sum-log_binary6451.9

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

  7. Applied add-cube-cbrt_binary6451.9

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

  8. Simplified51.9

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

  9. Simplified51.9

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

  10. Taylor expanded in hi around 0 50.4

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

  11. Final simplification50.4

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

Reproduce

herbie shell --seed 2022125 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))