xlohi (overflows)

Average Accuracy: 3.1% → 19.4%

Time: 14.1s

Precision: binary64

Cost: 20224

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (* (pow (cbrt (* (+ 1.0 (/ lo hi)) (- x lo))) 2.0) (/ (cbrt (- x 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 pow(cbrt(((1.0 + (lo / hi)) * (x - lo))), 2.0) * (cbrt((x - lo)) / hi);
}

Java

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

↓

public static double code(double lo, double hi, double x) {
	return Math.pow(Math.cbrt(((1.0 + (lo / hi)) * (x - lo))), 2.0) * (Math.cbrt((x - lo)) / hi);
}

Julia

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

↓

function code(lo, hi, x)
	return Float64((cbrt(Float64(Float64(1.0 + Float64(lo / hi)) * Float64(x - lo))) ^ 2.0) * Float64(cbrt(Float64(x - lo)) / hi))
end

Wolfram

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

↓

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

3. Simplified 9.3%

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

   [Start] 0.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   unpow2 [=>] 0.0	\[ \frac{lo \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 9.3	\[ \color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.3	\[ \frac{lo}{hi} \cdot \frac{x - lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]
   distribute-lft1-in [=>] 9.3	\[ \color{blue}{\left(\frac{lo}{hi} + 1\right) \cdot \frac{x - lo}{hi}} \]

4. Applied egg-rr 9.3%

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

   [Start] 9.3	\[ \left(\frac{lo}{hi} + 1\right) \cdot \frac{x - lo}{hi} \]
   associate-*r/ [=>] 9.3	\[ \color{blue}{\frac{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}{hi}} \]
   add-cube-cbrt [=>] 9.3	\[ \frac{\color{blue}{\left(\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}\right) \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}}{hi} \]
   associate-/l* [=>] 9.3	\[ \color{blue}{\frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{\frac{hi}{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}}} \]

5. Simplified 9.3%

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

   [Start] 9.3	\[ \frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{\frac{hi}{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}} \]
   associate-/r/ [=>] 9.3	\[ \color{blue}{\frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{hi} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}} \]
   *-commutative [=>] 9.3	\[ \color{blue}{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{hi}} \]
   *-commutative [=>] 9.3	\[ \sqrt[3]{\color{blue}{\left(x - lo\right) \cdot \left(\frac{lo}{hi} + 1\right)}} \cdot \frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{hi} \]
   +-commutative [=>] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \color{blue}{\left(1 + \frac{lo}{hi}\right)}} \cdot \frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)} \cdot \sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{hi} \]
   associate-/l* [=>] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \color{blue}{\frac{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}{\frac{hi}{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}}} \]
   *-commutative [=>] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(x - lo\right) \cdot \left(\frac{lo}{hi} + 1\right)}}}{\frac{hi}{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}} \]
   +-commutative [=>] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \frac{\sqrt[3]{\left(x - lo\right) \cdot \color{blue}{\left(1 + \frac{lo}{hi}\right)}}}{\frac{hi}{\sqrt[3]{\left(\frac{lo}{hi} + 1\right) \cdot \left(x - lo\right)}}} \]
   *-commutative [=>] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{\frac{hi}{\sqrt[3]{\color{blue}{\left(x - lo\right) \cdot \left(\frac{lo}{hi} + 1\right)}}}} \]
   +-commutative [=>] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{\frac{hi}{\sqrt[3]{\left(x - lo\right) \cdot \color{blue}{\left(1 + \frac{lo}{hi}\right)}}}} \]

6. Applied egg-rr 9.3%

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

   [Start] 9.3	\[ \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{\frac{hi}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}} \]
   associate-*r/ [=>] 9.3	\[ \color{blue}{\frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{\frac{hi}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}} \]
   div-inv [=>] 9.3	\[ \frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{\color{blue}{hi \cdot \frac{1}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}} \]
   associate-/r* [=>] 9.3	\[ \color{blue}{\frac{\frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)} \cdot \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{hi}}{\frac{1}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}} \]
   associate-*l/ [<=] 9.3	\[ \frac{\color{blue}{\frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{hi} \cdot \sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}{\frac{1}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}} \]
   associate-/r/ [<=] 9.3	\[ \frac{\color{blue}{\frac{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}{\frac{hi}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}}}{\frac{1}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}} \]
   clear-num [=>] 9.3	\[ \frac{\color{blue}{\frac{1}{\frac{\frac{hi}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}}}{\frac{1}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}} \]
   associate-/l/ [=>] 9.3	\[ \color{blue}{\frac{1}{\frac{1}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}} \cdot \frac{\frac{hi}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}{\sqrt[3]{\left(x - lo\right) \cdot \left(1 + \frac{lo}{hi}\right)}}}} \]

7. Simplified 9.3%

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

   [Start] 9.3	\[ \frac{1}{\frac{1}{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}} \cdot \frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}}} \]
   associate-*l/ [=>] 9.3	\[ \frac{1}{\color{blue}{\frac{1 \cdot \frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}}}{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}}} \]
   associate-/l* [<=] 9.3	\[ \color{blue}{\frac{1 \cdot \sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}{1 \cdot \frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}}}} \]
   *-lft-identity [=>] 9.3	\[ \frac{\color{blue}{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}}{1 \cdot \frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}}} \]
   *-commutative [=>] 9.3	\[ \frac{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}{\color{blue}{\frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}} \cdot 1}} \]
   associate-/l/ [<=] 9.3	\[ \color{blue}{\frac{\frac{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}{1}}{\frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}}}} \]
   /-rgt-identity [=>] 9.3	\[ \frac{\color{blue}{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}}{\frac{hi}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}}} \]
   associate-/r/ [=>] 9.3	\[ \color{blue}{\frac{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}{hi} \cdot {\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2}} \]
   *-commutative [=>] 9.3	\[ \color{blue}{{\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2} \cdot \frac{\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}}{hi}} \]

8. Taylor expanded in hi around inf 19.4%

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

9. Simplified 19.4%

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

   [Start] 19.4	\[ {\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2} \cdot \frac{{\left(x - lo\right)}^{0.3333333333333333}}{hi} \]
   unpow1/3 [=>] 19.4	\[ {\left(\sqrt[3]{\left(1 + \frac{lo}{hi}\right) \cdot \left(x - lo\right)}\right)}^{2} \cdot \frac{\color{blue}{\sqrt[3]{x - lo}}}{hi} \]

10. Final simplification 19.4%

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

Alternatives

Alternative 1
Accuracy	19.3%
Cost	6592

\[\left|\frac{hi}{lo}\right| \]

Alternative 2
Accuracy	18.8%
Cost	320

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

Alternative 3
Accuracy	18.8%
Cost	256

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

Alternative 4
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023140 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))