Average Error: 62.0 → 52.0
Time: 29.1s
Precision: binary64
Cost: 20288
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\mathsf{fma}\left(\frac{x}{\sqrt[3]{hi}}, \sqrt[3]{\frac{1}{hi \cdot hi}}, \frac{\frac{x}{hi} - 1}{hi} \cdot lo\right) \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (fma (/ x (cbrt hi)) (cbrt (/ 1.0 (* hi hi))) (* (/ (- (/ x hi) 1.0) hi) lo)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return fma((x / cbrt(hi)), cbrt((1.0 / (hi * hi))), ((((x / hi) - 1.0) / hi) * lo));
}

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return fma(Float64(x / cbrt(hi)), cbrt(Float64(1.0 / Float64(hi * hi))), Float64(Float64(Float64(Float64(x / hi) - 1.0) / hi) * lo))
end

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

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

Error

Derivation

  1. Initial program 62.0

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

  2. Taylor expanded in lo around 0 52.0

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

  3. Simplified52.0

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

  4. Applied egg-rr52.0

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

Alternatives

Alternative 1
Error52.0
Cost256
\[-\frac{lo}{hi} \]
Alternative 2
Error52.0
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2023010 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))