Average Error: 62.0 → 51.5
Time: 3.4s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{lo}{hi}}\\ \frac{x}{hi} \cdot \left(\frac{lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right)\right) + \left(\frac{x - lo}{hi} - \sqrt{{\left({t_0}^{2} \cdot \left(t_0 \cdot {\left(\frac{lo}{hi}\right)}^{2}\right) + lo \cdot \left(\frac{lo}{hi} \cdot \frac{1}{hi}\right)\right)}^{2}}\right) \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (cbrt (/ lo hi))))
   (+
    (* (/ x hi) (* (/ lo hi) (+ (/ lo hi) 1.0)))
    (-
     (/ (- x lo) hi)
     (sqrt
      (pow
       (+
        (* (pow t_0 2.0) (* t_0 (pow (/ lo hi) 2.0)))
        (* lo (* (/ lo hi) (/ 1.0 hi))))
       2.0))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	double t_0 = cbrt((lo / hi));
	return ((x / hi) * ((lo / hi) * ((lo / hi) + 1.0))) + (((x - lo) / hi) - sqrt(pow(((pow(t_0, 2.0) * (t_0 * pow((lo / hi), 2.0))) + (lo * ((lo / hi) * (1.0 / hi)))), 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) {
	double t_0 = Math.cbrt((lo / hi));
	return ((x / hi) * ((lo / hi) * ((lo / hi) + 1.0))) + (((x - lo) / hi) - Math.sqrt(Math.pow(((Math.pow(t_0, 2.0) * (t_0 * Math.pow((lo / hi), 2.0))) + (lo * ((lo / hi) * (1.0 / hi)))), 2.0)));
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	t_0 = cbrt(Float64(lo / hi))
	return Float64(Float64(Float64(x / hi) * Float64(Float64(lo / hi) * Float64(Float64(lo / hi) + 1.0))) + Float64(Float64(Float64(x - lo) / hi) - sqrt((Float64(Float64((t_0 ^ 2.0) * Float64(t_0 * (Float64(lo / hi) ^ 2.0))) + Float64(lo * Float64(Float64(lo / hi) * Float64(1.0 / hi)))) ^ 2.0))))
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := Block[{t$95$0 = N[Power[N[(lo / hi), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[(x / hi), $MachinePrecision] * N[(N[(lo / hi), $MachinePrecision] * N[(N[(lo / hi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] - N[Sqrt[N[Power[N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(t$95$0 * N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(lo * N[(N[(lo / hi), $MachinePrecision] * N[(1.0 / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\frac{lo}{hi}}\\
\frac{x}{hi} \cdot \left(\frac{lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right)\right) + \left(\frac{x - lo}{hi} - \sqrt{{\left({t_0}^{2} \cdot \left(t_0 \cdot {\left(\frac{lo}{hi}\right)}^{2}\right) + lo \cdot \left(\frac{lo}{hi} \cdot \frac{1}{hi}\right)\right)}^{2}}\right)
\end{array}

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

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

  6. Applied egg-rr51.5

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

  7. Final simplification51.5

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

Reproduce

herbie shell --seed 2022150 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))