Average Error: 62.0 → 51.5
Time: 2.9s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\left({\left(\frac{1}{\sqrt[3]{\frac{lo}{hi}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{hi}{lo}}}{lo}\right) \cdot \left(hi - x\right) \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (* (* (pow (/ 1.0 (cbrt (/ lo hi))) 2.0) (/ (cbrt (/ hi lo)) lo)) (- hi x)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (pow((1.0 / cbrt((lo / hi))), 2.0) * (cbrt((hi / lo)) / lo)) * (hi - x);
}

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
public static double code(double lo, double hi, double x) {
	return (Math.pow((1.0 / Math.cbrt((lo / hi))), 2.0) * (Math.cbrt((hi / lo)) / lo)) * (hi - x);
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return Float64(Float64((Float64(1.0 / cbrt(Float64(lo / hi))) ^ 2.0) * Float64(cbrt(Float64(hi / lo)) / lo)) * Float64(hi - x))
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[(N[(N[Power[N[(1.0 / N[Power[N[(lo / hi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(hi / lo), $MachinePrecision], 1/3], $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] * N[(hi - x), $MachinePrecision]), $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
\left({\left(\frac{1}{\sqrt[3]{\frac{lo}{hi}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{hi}{lo}}}{lo}\right) \cdot \left(hi - x\right)

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 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

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

  4. Taylor expanded in lo around 0 64.0

    \[\leadsto \color{blue}{\frac{{hi}^{2} - hi \cdot x}{{lo}^{2}}} \]

  5. Simplified51.5

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

  6. Applied egg-rr51.5

    \[\leadsto \color{blue}{\left(\frac{{\left(\sqrt[3]{\frac{hi}{lo}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{hi}{lo}}}{lo}\right)} \cdot \left(hi - x\right) \]

  7. Applied egg-rr51.5

    \[\leadsto \left(\frac{{\color{blue}{\left(\frac{1}{\sqrt[3]{\frac{lo}{hi}}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{hi}{lo}}}{lo}\right) \cdot \left(hi - x\right) \]

  8. Final simplification51.5

    \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\frac{lo}{hi}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{hi}{lo}}}{lo}\right) \cdot \left(hi - x\right) \]

Reproduce

herbie shell --seed 2022151 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))