Average Error: 62.0 → 48.6
Time: 2.7s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\frac{1 + {\left(\frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}\right)}^{3}}{\mathsf{fma}\left(\frac{hi}{lo}, \mathsf{fma}\left(\frac{hi}{lo}, 1 + \frac{hi}{lo}, -1\right), 1\right)} - \frac{hi}{lo} \cdot \frac{x}{lo} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (-
  (/
   (+ 1.0 (pow (/ (fma hi (/ hi lo) hi) lo) 3.0))
   (fma (/ hi lo) (fma (/ hi lo) (+ 1.0 (/ hi lo)) -1.0) 1.0))
  (* (/ hi lo) (/ x lo))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return ((1.0 + pow((fma(hi, (hi / lo), hi) / lo), 3.0)) / fma((hi / lo), fma((hi / lo), (1.0 + (hi / lo)), -1.0), 1.0)) - ((hi / lo) * (x / lo));
}

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 + (Float64(fma(hi, Float64(hi / lo), hi) / lo) ^ 3.0)) / fma(Float64(hi / lo), fma(Float64(hi / lo), Float64(1.0 + Float64(hi / lo)), -1.0), 1.0)) - Float64(Float64(hi / lo) * Float64(x / lo)))
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[(N[(N[(1.0 + N[Power[N[(N[(hi * N[(hi / lo), $MachinePrecision] + hi), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi / lo), $MachinePrecision] * N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(hi / lo), $MachinePrecision] * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
\frac{1 + {\left(\frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}\right)}^{3}}{\mathsf{fma}\left(\frac{hi}{lo}, \mathsf{fma}\left(\frac{hi}{lo}, 1 + \frac{hi}{lo}, -1\right), 1\right)} - \frac{hi}{lo} \cdot \frac{x}{lo}

Error

Bits error versus lo

Bits error versus hi

Bits error versus x

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

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

  4. Taylor expanded in hi around inf 51.9

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

  5. Applied flip3-+_binary6451.9

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

  6. Simplified51.9

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

  7. Simplified51.9

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

  8. Taylor expanded in hi around 0 48.6

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

  9. Final simplification48.6

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

Reproduce

herbie shell --seed 2022129 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))