Average Error: 62.0 → 51.6
Time: 2.4s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[hi \cdot \frac{\frac{\frac{hi}{lo}}{{\left(\sqrt[3]{lo}\right)}^{2}}}{\sqrt[3]{lo}} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (* hi (/ (/ (/ hi lo) (pow (cbrt lo) 2.0)) (cbrt lo))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return hi * (((hi / lo) / pow(cbrt(lo), 2.0)) / cbrt(lo));
}

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 hi * (((hi / lo) / Math.pow(Math.cbrt(lo), 2.0)) / Math.cbrt(lo));
}

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

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

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

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 hi around inf 64.0

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

  5. Simplified51.6

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

  6. Applied egg-rr51.6

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

  7. Applied egg-rr51.6

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

  8. Final simplification51.6

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

Reproduce

herbie shell --seed 2022166 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))