Average Error: 62.0 → 51.5
Time: 3.8s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\frac{x}{hi} \cdot \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} - \sqrt{{\left(\frac{lo}{hi} \cdot {\left({\left(\sqrt[3]{\frac{lo}{hi}}\right)}^{2}\right)}^{3} + {\left(\frac{lo}{hi}\right)}^{2}\right)}^{2}}\right) \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (+
  (* (/ x hi) (* (/ lo hi) (/ lo hi)))
  (-
   (/ (- x lo) hi)
   (sqrt
    (pow
     (+ (* (/ lo hi) (pow (pow (cbrt (/ lo hi)) 2.0) 3.0)) (pow (/ lo hi) 2.0))
     2.0)))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return ((x / hi) * ((lo / hi) * (lo / hi))) + (((x - lo) / hi) - sqrt(pow((((lo / hi) * pow(pow(cbrt((lo / hi)), 2.0), 3.0)) + pow((lo / hi), 2.0)), 2.0)));
}

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 ((x / hi) * ((lo / hi) * (lo / hi))) + (((x - lo) / hi) - Math.sqrt(Math.pow((((lo / hi) * Math.pow(Math.pow(Math.cbrt((lo / hi)), 2.0), 3.0)) + Math.pow((lo / hi), 2.0)), 2.0)));
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return Float64(Float64(Float64(x / hi) * Float64(Float64(lo / hi) * Float64(lo / hi))) + Float64(Float64(Float64(x - lo) / hi) - sqrt((Float64(Float64(Float64(lo / hi) * ((cbrt(Float64(lo / hi)) ^ 2.0) ^ 3.0)) + (Float64(lo / hi) ^ 2.0)) ^ 2.0))))
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[(N[(N[(x / hi), $MachinePrecision] * N[(N[(lo / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] - N[Sqrt[N[Power[N[(N[(N[(lo / hi), $MachinePrecision] * N[Power[N[Power[N[Power[N[(lo / hi), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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

  4. Applied egg-rr51.5

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

  5. Applied egg-rr51.5

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

  6. Taylor expanded in lo around inf 51.5

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

  7. Final simplification51.5

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

Reproduce

herbie shell --seed 2022162 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))