xlohi (overflows)

Average Error: 62.0 → 51.5

Time: 10.8s

Precision: binary64

Cost: 33536

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (fma
   -1.0
   (* x (+ (/ 1.0 lo) (/ hi (* lo lo))))
   (* (/ hi lo) (sqrt (log (exp (pow (+ 1.0 (/ hi lo)) 2.0))))))))

C

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

↓

double code(double lo, double hi, double x) {
	return 1.0 + fma(-1.0, (x * ((1.0 / lo) + (hi / (lo * lo)))), ((hi / lo) * sqrt(log(exp(pow((1.0 + (hi / lo)), 2.0))))));
}

Julia

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

↓

function code(lo, hi, x)
	return Float64(1.0 + fma(-1.0, Float64(x * Float64(Float64(1.0 / lo) + Float64(hi / Float64(lo * lo)))), Float64(Float64(hi / lo) * sqrt(log(exp((Float64(1.0 + Float64(hi / lo)) ^ 2.0)))))))
end

Wolfram

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

↓

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

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. Simplified 51.9

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

   [Start] 64.0	\[ \left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + -1 \cdot \frac{x}{lo}\right)} - -1 \cdot \frac{hi}{lo} \]
   associate--l+ [=>] 64.0	\[ \color{blue}{\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right)} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)} + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\color{blue}{\frac{-1 \cdot x}{lo}} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\frac{-1 \cdot x}{lo} - \color{blue}{\frac{-1 \cdot hi}{lo}}\right) \]
   div-sub [<=] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{\frac{-1 \cdot x - -1 \cdot hi}{lo}} \]
   distribute-lft-out-- [=>] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \frac{\color{blue}{-1 \cdot \left(x - hi\right)}}{lo} \]
   associate-*r/ [<=] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{-1 \cdot \frac{x - hi}{lo}} \]
   associate-+r+ [<=] 64.0	\[ \color{blue}{1 + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + -1 \cdot \frac{x - hi}{lo}\right)} \]

4. Taylor expanded in x around -inf 51.9

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

5. Simplified 51.9

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

   [Start] 51.9	\[ 1 + \left(-1 \cdot \left(\left(\frac{hi}{{lo}^{2}} + \frac{1}{lo}\right) \cdot x\right) + \frac{\left(1 + \frac{hi}{lo}\right) \cdot hi}{lo}\right) \]
   fma-def [=>] 51.9	\[ 1 + \color{blue}{\mathsf{fma}\left(-1, \left(\frac{hi}{{lo}^{2}} + \frac{1}{lo}\right) \cdot x, \frac{\left(1 + \frac{hi}{lo}\right) \cdot hi}{lo}\right)} \]
   *-commutative [=>] 51.9	\[ 1 + \mathsf{fma}\left(-1, \color{blue}{x \cdot \left(\frac{hi}{{lo}^{2}} + \frac{1}{lo}\right)}, \frac{\left(1 + \frac{hi}{lo}\right) \cdot hi}{lo}\right) \]
   +-commutative [=>] 51.9	\[ 1 + \mathsf{fma}\left(-1, x \cdot \color{blue}{\left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)}, \frac{\left(1 + \frac{hi}{lo}\right) \cdot hi}{lo}\right) \]
   unpow2 [=>] 51.9	\[ 1 + \mathsf{fma}\left(-1, x \cdot \left(\frac{1}{lo} + \frac{hi}{\color{blue}{lo \cdot lo}}\right), \frac{\left(1 + \frac{hi}{lo}\right) \cdot hi}{lo}\right) \]
   associate-*r/ [<=] 51.9	\[ 1 + \mathsf{fma}\left(-1, x \cdot \left(\frac{1}{lo} + \frac{hi}{lo \cdot lo}\right), \color{blue}{\left(1 + \frac{hi}{lo}\right) \cdot \frac{hi}{lo}}\right) \]
   *-commutative [=>] 51.9	\[ 1 + \mathsf{fma}\left(-1, x \cdot \left(\frac{1}{lo} + \frac{hi}{lo \cdot lo}\right), \color{blue}{\frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right)}\right) \]

6. Applied egg-rr 51.5

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

7. Applied egg-rr 51.5

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

8. Final simplification 51.5

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

Alternatives

Alternative 1
Error	51.5
Cost	14656

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

Alternative 2
Error	51.5
Cost	13824

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

Alternative 3
Error	51.5
Cost	13696

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

Alternative 4
Error	51.5
Cost	576

\[\frac{1}{\frac{lo}{hi} \cdot \frac{lo}{hi}} \]

Alternative 5
Error	51.5
Cost	448

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

Alternative 6
Error	52.1
Cost	320

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

Alternative 7
Error	52.0
Cost	320

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

Alternative 8
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023237 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))