Average Error: 62.0 → 50.5
Time: 3.6s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := 1 - \frac{x}{lo}\\ t_1 := {\left(\frac{1}{{t_0}^{2}}\right)}^{0.05555555555555555}\\ 1 + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(\left(\frac{1}{lo} - \frac{x}{lo \cdot lo}\right) \cdot hi\right) \cdot \left(t_1 \cdot t_1\right), \sqrt[3]{t_0}\right), \sqrt[3]{{\left(\mathsf{fma}\left(-1 - \frac{hi}{lo}, \frac{x - hi}{lo}, 1\right)\right)}^{2}}, -1\right) \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x lo)))
        (t_1 (pow (/ 1.0 (pow t_0 2.0)) 0.05555555555555555)))
   (+
    1.0
    (fma
     (fma
      0.3333333333333333
      (* (* (- (/ 1.0 lo) (/ x (* lo lo))) hi) (* t_1 t_1))
      (cbrt t_0))
     (cbrt (pow (fma (- -1.0 (/ hi lo)) (/ (- x hi) lo) 1.0) 2.0))
     -1.0))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	double t_0 = 1.0 - (x / lo);
	double t_1 = pow((1.0 / pow(t_0, 2.0)), 0.05555555555555555);
	return 1.0 + fma(fma(0.3333333333333333, ((((1.0 / lo) - (x / (lo * lo))) * hi) * (t_1 * t_1)), cbrt(t_0)), cbrt(pow(fma((-1.0 - (hi / lo)), ((x - hi) / lo), 1.0), 2.0)), -1.0);
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	t_0 = Float64(1.0 - Float64(x / lo))
	t_1 = Float64(1.0 / (t_0 ^ 2.0)) ^ 0.05555555555555555
	return Float64(1.0 + fma(fma(0.3333333333333333, Float64(Float64(Float64(Float64(1.0 / lo) - Float64(x / Float64(lo * lo))) * hi) * Float64(t_1 * t_1)), cbrt(t_0)), cbrt((fma(Float64(-1.0 - Float64(hi / lo)), Float64(Float64(x - hi) / lo), 1.0) ^ 2.0)), -1.0))
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 0.05555555555555555], $MachinePrecision]}, N[(1.0 + N[(N[(0.3333333333333333 * N[(N[(N[(N[(1.0 / lo), $MachinePrecision] - N[(x / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * hi), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] + 1.0), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := 1 - \frac{x}{lo}\\
t_1 := {\left(\frac{1}{{t_0}^{2}}\right)}^{0.05555555555555555}\\
1 + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(\left(\frac{1}{lo} - \frac{x}{lo \cdot lo}\right) \cdot hi\right) \cdot \left(t_1 \cdot t_1\right), \sqrt[3]{t_0}\right), \sqrt[3]{{\left(\mathsf{fma}\left(-1 - \frac{hi}{lo}, \frac{x - hi}{lo}, 1\right)\right)}^{2}}, -1\right)
\end{array}

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

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

  4. Applied egg-rr51.9

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

  5. Taylor expanded in hi around 0 50.5

    \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{0.3333333333333333 \cdot \left(\left(hi \cdot \left(\frac{1}{lo} + -1 \cdot \frac{x}{{lo}^{2}}\right)\right) \cdot {\left(\frac{1}{{\left(-1 \cdot \frac{x}{lo} + 1\right)}^{2}}\right)}^{0.1111111111111111}\right) + {\left(-1 \cdot \frac{x}{lo} + 1\right)}^{0.3333333333333333}}, \sqrt[3]{{\left(\mathsf{fma}\left(-1 - \frac{hi}{lo}, \frac{x - hi}{lo}, 1\right)\right)}^{2}}, -1\right) \]

  6. Simplified50.5

    \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, \left(\left(\frac{1}{lo} - \frac{x}{lo \cdot lo}\right) \cdot hi\right) \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.05555555555555555} \cdot {\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.05555555555555555}\right), \sqrt[3]{1 - \frac{x}{lo}}\right)}, \sqrt[3]{{\left(\mathsf{fma}\left(-1 - \frac{hi}{lo}, \frac{x - hi}{lo}, 1\right)\right)}^{2}}, -1\right) \]

  7. Final simplification50.5

    \[\leadsto 1 + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(\left(\frac{1}{lo} - \frac{x}{lo \cdot lo}\right) \cdot hi\right) \cdot \left({\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.05555555555555555} \cdot {\left(\frac{1}{{\left(1 - \frac{x}{lo}\right)}^{2}}\right)}^{0.05555555555555555}\right), \sqrt[3]{1 - \frac{x}{lo}}\right), \sqrt[3]{{\left(\mathsf{fma}\left(-1 - \frac{hi}{lo}, \frac{x - hi}{lo}, 1\right)\right)}^{2}}, -1\right) \]

Reproduce

herbie shell --seed 2022206 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))