Average Error: 62.0 → 50.4
Time: 3.6s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + \frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right)}\\ t_0 \cdot \left(t_0 \cdot \mathsf{fma}\left(\frac{hi}{lo}, 0.3333333333333333, 1\right)\right) - \frac{hi}{lo} \cdot \frac{x}{lo} \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{1 + \frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right)}\\
t_0 \cdot \left(t_0 \cdot \mathsf{fma}\left(\frac{hi}{lo}, 0.3333333333333333, 1\right)\right) - \frac{hi}{lo} \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 (cbrt (+ 1.0 (* (/ hi lo) (+ 1.0 (/ hi lo)))))))
   (-
    (* t_0 (* t_0 (fma (/ hi lo) 0.3333333333333333 1.0)))
    (* (/ 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(1.0 + ((hi / lo) * (1.0 + (hi / lo))));
	return (t_0 * (t_0 * fma((hi / lo), 0.3333333333333333, 1.0))) - ((hi / lo) * (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. Taylor expanded in hi around inf 51.9

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

  5. Applied add-cube-cbrt_binary6451.9

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

  6. Taylor expanded in hi around 0 50.4

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

  7. Simplified50.4

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

  8. Final simplification50.4

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

Reproduce

herbie shell --seed 2021313 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))