Average Error: 62.0 → 52.1
Time: 4.1s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\sqrt[3]{1 - \frac{x}{lo}} \cdot \sqrt[3]{{\left(\sqrt[3]{1 - \frac{x}{lo}}\right)}^{6}}\]
\frac{x - lo}{hi - lo}
↓
\sqrt[3]{1 - \frac{x}{lo}} \cdot \sqrt[3]{{\left(\sqrt[3]{1 - \frac{x}{lo}}\right)}^{6}}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (* (cbrt (- 1.0 (/ x lo))) (cbrt (pow (cbrt (- 1.0 (/ x lo))) 6.0))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return cbrt(1.0 - (x / lo)) * cbrt(pow(cbrt(1.0 - (x / lo)), 6.0));
}

Error

Bits error versus lo

Bits error versus hi

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 62.0

    \[\frac{x - lo}{hi - lo}\]

  2. Taylor expanded around 0 52.1

    \[\leadsto \color{blue}{-1 \cdot \frac{x - lo}{lo}}\]

  3. Simplified52.1

    \[\leadsto \color{blue}{1 - \frac{x}{lo}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt_binary64_79552.1

    \[\leadsto \color{blue}{\left(\sqrt[3]{1 - \frac{x}{lo}} \cdot \sqrt[3]{1 - \frac{x}{lo}}\right) \cdot \sqrt[3]{1 - \frac{x}{lo}}}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube_binary64_79652.1

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

  8. Simplified52.1

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{1 - \frac{x}{lo}}\right)}^{6}}} \cdot \sqrt[3]{1 - \frac{x}{lo}}\]

  9. Final simplification52.1

    \[\leadsto \sqrt[3]{1 - \frac{x}{lo}} \cdot \sqrt[3]{{\left(\sqrt[3]{1 - \frac{x}{lo}}\right)}^{6}}\]

Reproduce

herbie shell --seed 2021024 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))