xlohi (overflows)

Average Accuracy: 3.1% → 20.6%

Time: 14.3s

Precision: binary64

Cost: 20416

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (log
  (fma (+ (/ x (* hi hi)) (/ -1.0 hi)) (+ lo (/ lo (/ hi x))) (exp (/ x hi)))))

C

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

↓

double code(double lo, double hi, double x) {
	return log(fma(((x / (hi * hi)) + (-1.0 / hi)), (lo + (lo / (hi / x))), exp((x / hi))));
}

Julia

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

↓

function code(lo, hi, x)
	return log(fma(Float64(Float64(x / Float64(hi * hi)) + Float64(-1.0 / hi)), Float64(lo + Float64(lo / Float64(hi / x))), exp(Float64(x / hi))))
end

Wolfram

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

↓

code[lo_, hi_, x_] := N[Log[N[(N[(N[(x / N[(hi * hi), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / hi), $MachinePrecision]), $MachinePrecision] * N[(lo + N[(lo / N[(hi / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x / hi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

3. Simplified 18.8%

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

   [Start] 18.8	\[ \frac{x}{hi} + -1 \cdot \left(lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right) \]
   mul-1-neg [=>] 18.8	\[ \frac{x}{hi} + \color{blue}{\left(-lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right)} \]
   unsub-neg [=>] 18.8	\[ \color{blue}{\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)} \]
   mul-1-neg [=>] 18.8	\[ \frac{x}{hi} - lo \cdot \left(\frac{1}{hi} + \color{blue}{\left(-\frac{x}{{hi}^{2}}\right)}\right) \]
   unsub-neg [=>] 18.8	\[ \frac{x}{hi} - lo \cdot \color{blue}{\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)} \]
   unpow2 [=>] 18.8	\[ \frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \frac{x}{\color{blue}{hi \cdot hi}}\right) \]

4. Applied egg-rr 18.8%

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

   [Start] 18.8	\[ \frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right) \]
   add-log-exp [=>] 18.8	\[ \color{blue}{\log \left(e^{\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)}\right)} \]
   associate-/r* [=>] 18.8	\[ \log \left(e^{\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \color{blue}{\frac{\frac{x}{hi}}{hi}}\right)}\right) \]
   sub-div [=>] 18.8	\[ \log \left(e^{\frac{x}{hi} - lo \cdot \color{blue}{\frac{1 - \frac{x}{hi}}{hi}}}\right) \]

5. Taylor expanded in lo around 0 20.6%

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

6. Simplified 20.6%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{x}{hi \cdot hi} - \frac{1}{hi}, lo \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)} \]
   Proof

   [Start] 20.6	\[ \log \left(\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot \left(lo \cdot e^{\frac{x}{hi}}\right) + e^{\frac{x}{hi}}\right) \]
   fma-def [=>] 20.6	\[ \log \color{blue}{\left(\mathsf{fma}\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}, lo \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right)} \]
   unpow2 [=>] 20.6	\[ \log \left(\mathsf{fma}\left(\frac{x}{\color{blue}{hi \cdot hi}} - \frac{1}{hi}, lo \cdot e^{\frac{x}{hi}}, e^{\frac{x}{hi}}\right)\right) \]

7. Taylor expanded in x around 0 10.2%

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

8. Simplified 20.6%

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

   [Start] 10.2	\[ \log \left(\mathsf{fma}\left(\frac{x}{hi \cdot hi} - \frac{1}{hi}, lo + \frac{lo \cdot x}{hi}, e^{\frac{x}{hi}}\right)\right) \]
   associate-/l* [=>] 20.6	\[ \log \left(\mathsf{fma}\left(\frac{x}{hi \cdot hi} - \frac{1}{hi}, lo + \color{blue}{\frac{lo}{\frac{hi}{x}}}, e^{\frac{x}{hi}}\right)\right) \]

9. Final simplification 20.6%

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

Alternatives

Alternative 1
Accuracy	20.6%
Cost	13760

\[\log \left(e^{\frac{x}{hi}} \cdot \left(1 + lo \cdot \frac{\frac{x}{hi} + -1}{hi}\right)\right) \]

Alternative 2
Accuracy	20.6%
Cost	6848

\[\log \left(1 + \frac{x - lo}{hi}\right) \]

Alternative 3
Accuracy	20.6%
Cost	6656

\[\mathsf{log1p}\left(\frac{-lo}{hi}\right) \]

Alternative 4
Accuracy	20.2%
Cost	1344

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

Alternative 5
Accuracy	18.9%
Cost	832

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

Alternative 6
Accuracy	18.9%
Cost	704

\[1 + \frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right) \]

Alternative 7
Accuracy	18.8%
Cost	320

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

Alternative 8
Accuracy	18.8%
Cost	256

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

Alternative 9
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023139 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))