xlohi (overflows)

Average Error: 62.0 → 34.9

Time: 14.5s

Precision: binary64

Cost: 7936

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

Math

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

↓

\[\frac{1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{6}}{\left(1 + \frac{x}{lo}\right) - \frac{hi}{lo}} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (/
  (- 1.0 (pow (* (/ (- hi x) lo) (+ 1.0 (/ hi lo))) 6.0))
  (- (+ 1.0 (/ x lo)) (/ hi 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 - pow((((hi - x) / lo) * (1.0 + (hi / lo))), 6.0)) / ((1.0 + (x / lo)) - (hi / 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) * (1.0d0 + (hi / lo))) ** 6.0d0)) / ((1.0d0 + (x / lo)) - (hi / 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 - Math.pow((((hi - x) / lo) * (1.0 + (hi / lo))), 6.0)) / ((1.0 + (x / lo)) - (hi / lo));
}

Python

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

↓

def code(lo, hi, x):
	return (1.0 - math.pow((((hi - x) / lo) * (1.0 + (hi / lo))), 6.0)) / ((1.0 + (x / lo)) - (hi / lo))

Julia

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

↓

function code(lo, hi, x)
	return Float64(Float64(1.0 - (Float64(Float64(Float64(hi - x) / lo) * Float64(1.0 + Float64(hi / lo))) ^ 6.0)) / Float64(Float64(1.0 + Float64(x / lo)) - Float64(hi / 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) * (1.0 + (hi / lo))) ^ 6.0)) / ((1.0 + (x / lo)) - (hi / lo));
end

Wolfram

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

↓

code[lo_, hi_, x_] := N[(N[(1.0 - N[Power[N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(x / lo), $MachinePrecision]), $MachinePrecision] - N[(hi / lo), $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. Applied egg-rr 51.9

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

5. Simplified 51.9

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{6}}{\left(1 - \frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right) \cdot \left(1 + \left({\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{2} + {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{4}\right)\right)}} \]
   Proof

   [Start] 51.9

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

6. Taylor expanded in hi around 0 46.7

   \[\leadsto \frac{1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{6}}{\left(1 - \frac{hi - x}{lo} \cdot \color{blue}{1}\right) \cdot \left(1 + \left({\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{2} + {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{4}\right)\right)} \]

7. Taylor expanded in lo around inf 34.9

   \[\leadsto \frac{1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{6}}{\color{blue}{\left(1 + \frac{x}{lo}\right) - \frac{hi}{lo}}} \]

8. Final simplification 34.9

   \[\leadsto \frac{1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{6}}{\left(1 + \frac{x}{lo}\right) - \frac{hi}{lo}} \]

Alternatives

Alternative 1
Error	34.9
Cost	7808

\[\frac{1 - {\left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{lo}\right)\right)}^{6}}{1 + \frac{x - hi}{lo}} \]

Alternative 2
Error	51.5
Cost	448

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

Alternative 3
Error	52.0
Cost	256

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

Alternative 4
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023053 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))