Average Error: 62.0 → 51.9
Time: 2.7s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[1 + \mathsf{fma}\left(\frac{\sqrt{hi}}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{lo}\right)}^{2}\right)} + -1}, \frac{\sqrt{hi}}{\sqrt[3]{lo}}, 1\right) \cdot \sqrt{{\left(\frac{hi - x}{lo}\right)}^{2}} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (*
   (fma
    (/ (sqrt hi) (+ (exp (log1p (pow (cbrt lo) 2.0))) -1.0))
    (/ (sqrt hi) (cbrt lo))
    1.0)
   (sqrt (pow (/ (- hi x) 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 1.0 + (fma((sqrt(hi) / (exp(log1p(pow(cbrt(lo), 2.0))) + -1.0)), (sqrt(hi) / cbrt(lo)), 1.0) * sqrt(pow(((hi - x) / lo), 2.0)));
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return Float64(1.0 + Float64(fma(Float64(sqrt(hi) / Float64(exp(log1p((cbrt(lo) ^ 2.0))) + -1.0)), Float64(sqrt(hi) / cbrt(lo)), 1.0) * sqrt((Float64(Float64(hi - x) / lo) ^ 2.0))))
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(N[Sqrt[hi], $MachinePrecision] / N[(N[Exp[N[Log[1 + N[Power[N[Power[lo, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[hi], $MachinePrecision] / N[Power[lo, 1/3], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
1 + \mathsf{fma}\left(\frac{\sqrt{hi}}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{lo}\right)}^{2}\right)} + -1}, \frac{\sqrt{hi}}{\sqrt[3]{lo}}, 1\right) \cdot \sqrt{{\left(\frac{hi - x}{lo}\right)}^{2}}

Error

Derivation

  1. Initial program 62.0

    \[\frac{x - lo}{hi - lo} \]

  2. Taylor expanded in lo around inf 64.0

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

  3. Simplified51.9

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

  4. Applied egg-rr51.9

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

  5. Applied egg-rr51.9

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

  6. Applied egg-rr51.9

    \[\leadsto 1 + \mathsf{fma}\left(\frac{\sqrt{hi}}{\color{blue}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{lo}\right)}^{2}\right)} - 1}}, \frac{\sqrt{hi}}{\sqrt[3]{lo}}, 1\right) \cdot \sqrt{{\left(\frac{hi - x}{lo}\right)}^{2}} \]

  7. Final simplification51.9

    \[\leadsto 1 + \mathsf{fma}\left(\frac{\sqrt{hi}}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{lo}\right)}^{2}\right)} + -1}, \frac{\sqrt{hi}}{\sqrt[3]{lo}}, 1\right) \cdot \sqrt{{\left(\frac{hi - x}{lo}\right)}^{2}} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))