Average Error: 62.0 → 51.9
Time: 3.5s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\frac{hi}{lo} + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(2 + {\left(\frac{hi}{lo}\right)}^{2}\right)\right)\right)\right) \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (+ (/ hi lo) (expm1 (log1p (expm1 (log (+ 2.0 (pow (/ hi lo) 2.0))))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (hi / lo) + expm1(log1p(expm1(log((2.0 + pow((hi / lo), 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 (hi / lo) + Math.expm1(Math.log1p(Math.expm1(Math.log((2.0 + Math.pow((hi / lo), 2.0))))));
}

def code(lo, hi, x):
	return (x - lo) / (hi - lo)
↓
def code(lo, hi, x):
	return (hi / lo) + math.expm1(math.log1p(math.expm1(math.log((2.0 + math.pow((hi / lo), 2.0))))))

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return Float64(Float64(hi / lo) + expm1(log1p(expm1(log(Float64(2.0 + (Float64(hi / lo) ^ 2.0)))))))
end

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

\frac{x - lo}{hi - lo}
↓
\frac{hi}{lo} + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(2 + {\left(\frac{hi}{lo}\right)}^{2}\right)\right)\right)\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 x around 0 64.0

    \[\leadsto \frac{hi}{lo} + \color{blue}{\left(1 + \frac{{hi}^{2}}{{lo}^{2}}\right)} \]

  5. Simplified51.9

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

  6. Applied egg-rr51.9

    \[\leadsto \frac{hi}{lo} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + {\left(\frac{hi}{lo}\right)}^{2}\right)\right)} \]

  7. Applied egg-rr51.9

    \[\leadsto \frac{hi}{lo} + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(2 + {\left(\frac{hi}{lo}\right)}^{2}\right)\right)}\right)\right) \]

  8. Final simplification51.9

    \[\leadsto \frac{hi}{lo} + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(2 + {\left(\frac{hi}{lo}\right)}^{2}\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022156 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))