xlohi (overflows)

Average Error: 62.0 → 1.0

Time: 12.1s

Precision: binary64

Cost: 21184

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

↓

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

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

↓

(FPCore (lo hi x)
 :precision binary64
 (+
  (/ x hi)
  (/
   (- (pow (* (- x lo) (/ lo (* hi hi))) 2.0) (pow (/ lo hi) 2.0))
   (fma (/ lo hi) (/ (- x lo) hi) (/ lo hi)))))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

↓

double code(double lo, double hi, double x) {
	return (x / hi) + ((pow(((x - lo) * (lo / (hi * hi))), 2.0) - pow((lo / hi), 2.0)) / fma((lo / hi), ((x - lo) / hi), (lo / hi)));
}

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

↓

function code(lo, hi, x)
	return Float64(Float64(x / hi) + Float64(Float64((Float64(Float64(x - lo) * Float64(lo / Float64(hi * hi))) ^ 2.0) - (Float64(lo / hi) ^ 2.0)) / fma(Float64(lo / hi), Float64(Float64(x - lo) / hi), Float64(lo / hi))))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

↓

code[lo_, hi_, x_] := N[(N[(x / hi), $MachinePrecision] + N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] * N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(lo / hi), $MachinePrecision] * N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] + N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0

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

2. Taylor expanded in hi around inf 64.0

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

3. Simplified 57.9

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

   [Start] 64.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 64.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   unpow2 [=>] 64.0	\[ \frac{lo \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 57.9	\[ \color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 57.9	\[ \frac{lo}{hi} \cdot \frac{x - lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]
   distribute-lft1-in [=>] 57.9	\[ \color{blue}{\left(\frac{lo}{hi} + 1\right) \cdot \frac{x - lo}{hi}} \]

4. Applied egg-rr 57.9

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

5. Applied egg-rr 1.0

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

6. Final simplification 1.0

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

Alternatives

Alternative 1
Error	51.5
Cost	20224

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

Alternative 2
Error	51.5
Cost	7744

\[\begin{array}{l} t_0 := 1 + \frac{hi}{lo}\\ \left(2 + \frac{hi - x}{lo} \cdot \sqrt{t_0 \cdot t_0}\right) + -1 \end{array} \]

Alternative 3
Error	51.6
Cost	576

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

Alternative 4
Error	51.6
Cost	448

\[\frac{hi}{lo} \cdot \frac{hi}{lo} \]

Alternative 5
Error	52.0
Cost	256

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

Alternative 6
Error	52.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022349 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))