Average Error: 62.0 → 51.5
Time: 3.5s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{hi}{lo}}\\ t_1 := 1 + \frac{hi}{lo}\\ \left(1 + \frac{hi}{lo} \cdot \sqrt{\mathsf{fma}\left(hi, \frac{1}{lo} \cdot \mathsf{fma}\left({t_0}^{2}, t_0, 1\right), t_1\right)}\right) - t_1 \cdot \frac{x}{lo} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (cbrt (/ hi lo))) (t_1 (+ 1.0 (/ hi lo))))
   (-
    (+
     1.0
     (*
      (/ hi lo)
      (sqrt (fma hi (* (/ 1.0 lo) (fma (pow t_0 2.0) t_0 1.0)) t_1))))
    (* t_1 (/ x lo)))))
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((hi / lo));
	double t_1 = 1.0 + (hi / lo);
	return (1.0 + ((hi / lo) * sqrt(fma(hi, ((1.0 / lo) * fma(pow(t_0, 2.0), t_0, 1.0)), t_1)))) - (t_1 * (x / lo));
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	t_0 = cbrt(Float64(hi / lo))
	t_1 = Float64(1.0 + Float64(hi / lo))
	return Float64(Float64(1.0 + Float64(Float64(hi / lo) * sqrt(fma(hi, Float64(Float64(1.0 / lo) * fma((t_0 ^ 2.0), t_0, 1.0)), t_1)))) - Float64(t_1 * Float64(x / lo)))
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[(hi / lo), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[Sqrt[N[(hi * N[(N[(1.0 / lo), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\frac{hi}{lo}}\\
t_1 := 1 + \frac{hi}{lo}\\
\left(1 + \frac{hi}{lo} \cdot \sqrt{\mathsf{fma}\left(hi, \frac{1}{lo} \cdot \mathsf{fma}\left({t_0}^{2}, t_0, 1\right), t_1\right)}\right) - t_1 \cdot \frac{x}{lo}
\end{array}

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. Applied egg-rr51.5

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

  5. Applied egg-rr51.5

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

  6. Applied egg-rr51.5

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

  7. Final simplification51.5

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

Reproduce

herbie shell --seed 2022146 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))