xlohi (overflows)

Average Error: 96.87% → 0.59%

Time: 11.0s

Precision: binary64

Cost: 832

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

Math

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

↓

\[\frac{1}{\frac{hi}{x - lo} - \frac{lo}{x - lo}} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (/ 1.0 (- (/ hi (- x lo)) (/ lo (- x lo)))))

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 / ((hi / (x - lo)) - (lo / (x - lo)));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function

↓

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 / ((hi / (x - lo)) - (lo / (x - lo)))
end function

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 1.0 / ((hi / (x - lo)) - (lo / (x - lo)));
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

↓

def code(lo, hi, x):
	return 1.0 / ((hi / (x - lo)) - (lo / (x - lo)))

Julia

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

↓

function code(lo, hi, x)
	return Float64(1.0 / Float64(Float64(hi / Float64(x - lo)) - Float64(lo / Float64(x - lo))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

↓

function tmp = code(lo, hi, x)
	tmp = 1.0 / ((hi / (x - lo)) - (lo / (x - lo)));
end

Wolfram

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

↓

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

Derivation

1. Initial program 96.87

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

2. Taylor expanded in hi around inf 100

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

3. Simplified 90.51

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

   [Start] 100	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 100	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 100	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   unpow2 [=>] 100	\[ \frac{lo \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 90.51	\[ \color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 90.51	\[ \frac{lo}{hi} \cdot \frac{x - lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]
   distribute-lft1-in [=>] 90.51	\[ \color{blue}{\left(\frac{lo}{hi} + 1\right) \cdot \frac{x - lo}{hi}} \]

4. Applied egg-rr 90.51

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

5. Taylor expanded in hi around inf 0.59

   \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo} + -1 \cdot \frac{lo}{x - lo}}} \]

6. Simplified 0.59

   \[\leadsto \frac{1}{\color{blue}{\frac{hi}{x - lo} + \frac{-lo}{x - lo}}} \]
   Proof

   [Start] 0.59	\[ \frac{1}{\frac{hi}{x - lo} + -1 \cdot \frac{lo}{x - lo}} \]
   associate-*r/ [=>] 0.59	\[ \frac{1}{\frac{hi}{x - lo} + \color{blue}{\frac{-1 \cdot lo}{x - lo}}} \]
   neg-mul-1 [<=] 0.59	\[ \frac{1}{\frac{hi}{x - lo} + \frac{\color{blue}{-lo}}{x - lo}} \]

7. Final simplification 0.59

   \[\leadsto \frac{1}{\frac{hi}{x - lo} - \frac{lo}{x - lo}} \]

Alternatives

Alternative 1
Error	1.01%
Cost	832

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

Alternative 2
Error	1.6%
Cost	448

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

Alternative 3
Error	81.21%
Cost	320

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

Alternative 4
Error	81.2%
Cost	256

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

Alternative 5
Error	81.33%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023103 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))