Average Error: 62.0 → 51.9
Time: 7.7s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log \left(\sqrt[3]{lo} \cdot \sqrt[3]{lo}\right)}\\ \frac{x}{hi} + \left(\mathsf{fma}\left(\frac{x}{hi}, \frac{lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right), \frac{lo}{hi} \cdot \left(-1 - \frac{lo}{hi}\right)\right) - e^{\mathsf{fma}\left(t_0 \cdot t_0, t_0, -\log \left(\sqrt[3]{hi} \cdot \sqrt[3]{hi}\right)\right) \cdot 3} \cdot {\left(\frac{\sqrt[3]{lo}}{\sqrt[3]{hi}}\right)}^{3}\right) \end{array} \]
\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\log \left(\sqrt[3]{lo} \cdot \sqrt[3]{lo}\right)}\\
\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) - e^{\mathsf{fma}\left(t_0 \cdot t_0, t_0, -\log \left(\sqrt[3]{hi} \cdot \sqrt[3]{hi}\right)\right) \cdot 3} \cdot {\left(\frac{\sqrt[3]{lo}}{\sqrt[3]{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 (* (cbrt lo) (cbrt lo))))))
   (+
    (/ x hi)
    (-
     (fma
      (/ x hi)
      (* (/ lo hi) (+ (/ lo hi) 1.0))
      (* (/ lo hi) (- -1.0 (/ lo hi))))
     (*
      (exp (* (fma (* t_0 t_0) t_0 (- (log (* (cbrt hi) (cbrt hi))))) 3.0))
      (pow (/ (cbrt lo) (cbrt 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(cbrt(lo) * cbrt(lo)));
	return (x / hi) + (fma((x / hi), ((lo / hi) * ((lo / hi) + 1.0)), ((lo / hi) * (-1.0 - (lo / hi)))) - (exp(fma((t_0 * t_0), t_0, -log(cbrt(hi) * cbrt(hi))) * 3.0) * pow((cbrt(lo) / cbrt(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

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

  5. Applied add-cube-cbrt_binary6451.9

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

  6. Applied times-frac_binary6451.9

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

  7. Applied unpow-prod-down_binary6451.9

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

  8. Applied add-exp-log_binary6451.9

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

  9. Applied add-exp-log_binary6451.9

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

  10. Applied div-exp_binary6451.9

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

  11. Applied pow-exp_binary6451.9

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

  12. Applied add-cube-cbrt_binary6451.9

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

  13. Applied fma-neg_binary6451.9

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

  14. Final simplification51.9

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

Reproduce

herbie shell --seed 2022093 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))