Average Error: 62.0 → 52.0
Time: 4.3s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[{\left(x - lo\right)}^{0.6666666666666666} \cdot \frac{\sqrt[3]{x - lo}}{hi}\]
\frac{x - lo}{hi - lo}
↓
{\left(x - lo\right)}^{0.6666666666666666} \cdot \frac{\sqrt[3]{x - lo}}{hi}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (* (pow (- x lo) 0.6666666666666666) (/ (cbrt (- x lo)) hi)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return pow((x - lo), 0.6666666666666666) * (cbrt(x - lo) / hi);
}

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 inf 52.0

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

  4. Applied *-un-lft-identity_binary64_76052.0

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

  5. Applied add-cube-cbrt_binary64_79552.0

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

  6. Applied times-frac_binary64_76652.0

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi}}\]

  7. Simplified52.0

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

  9. Applied pow1/3_binary64_84252.0

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

  10. Applied pow1/3_binary64_84252.0

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

  11. Applied pow-prod-up_binary64_83052.0

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

  12. Final simplification52.0

    \[\leadsto {\left(x - lo\right)}^{0.6666666666666666} \cdot \frac{\sqrt[3]{x - lo}}{hi}\]

Reproduce

herbie shell --seed 2021059 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))