xlohi (overflows)

Average Error: 62.0 → 51.3

Time: 10.7s

Precision: binary64

Cost: 27264

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (*
   (* (/ 1.0 (+ 1.0 (- (pow (/ hi lo) 2.0) (/ hi lo)))) (pow (/ hi lo) 3.0))
   (sqrt (pow (/ (- hi x) 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 + (((1.0 / (1.0 + (pow((hi / lo), 2.0) - (hi / lo)))) * pow((hi / lo), 3.0)) * sqrt(pow(((hi - x) / lo), 2.0)));
}

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 + (((1.0d0 / (1.0d0 + (((hi / lo) ** 2.0d0) - (hi / lo)))) * ((hi / lo) ** 3.0d0)) * sqrt((((hi - x) / lo) ** 2.0d0)))
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 + (((1.0 / (1.0 + (Math.pow((hi / lo), 2.0) - (hi / lo)))) * Math.pow((hi / lo), 3.0)) * Math.sqrt(Math.pow(((hi - x) / lo), 2.0)));
}

Python

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

↓

def code(lo, hi, x):
	return 1.0 + (((1.0 / (1.0 + (math.pow((hi / lo), 2.0) - (hi / lo)))) * math.pow((hi / lo), 3.0)) * math.sqrt(math.pow(((hi - x) / 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 + Float64(Float64(Float64(1.0 / Float64(1.0 + Float64((Float64(hi / lo) ^ 2.0) - Float64(hi / lo)))) * (Float64(hi / lo) ^ 3.0)) * sqrt((Float64(Float64(hi - x) / lo) ^ 2.0))))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	tmp = 1.0 + (((1.0 / (1.0 + (((hi / lo) ^ 2.0) - (hi / lo)))) * ((hi / lo) ^ 3.0)) * sqrt((((hi - x) / 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[(N[(N[(1.0 / N[(1.0 + N[(N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision] - N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision], 2.0], $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. Applied egg-rr 51.9

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

5. Applied egg-rr 51.9

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

6. Taylor expanded in hi around inf 64.0

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

7. Simplified 51.3

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

   [Start] 64.0	\[ 1 + \left(\frac{1}{1 + \left({\left(\frac{hi}{lo}\right)}^{2} - \frac{hi}{lo}\right)} \cdot \frac{{hi}^{3}}{{lo}^{3}}\right) \cdot \sqrt{{\left(\frac{hi - x}{lo}\right)}^{2}} \]
   cube-div [<=] 51.3	\[ 1 + \left(\frac{1}{1 + \left({\left(\frac{hi}{lo}\right)}^{2} - \frac{hi}{lo}\right)} \cdot \color{blue}{{\left(\frac{hi}{lo}\right)}^{3}}\right) \cdot \sqrt{{\left(\frac{hi - x}{lo}\right)}^{2}} \]

8. Final simplification 51.3

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

Alternatives

Alternative 1
Error	51.5
Cost	7104

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

Alternative 2
Error	51.9
Cost	832

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

Alternative 3
Error	51.9
Cost	832

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

Alternative 4
Error	51.9
Cost	704

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

Alternative 5
Error	52.0
Cost	576

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

Alternative 6
Error	52.0
Cost	256

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

Alternative 7
Error	52.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023047 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))