xlohi (overflows)

Average Error: 96.87% → 0.95%

Time: 13.1s

Precision: binary64

Cost: 13824

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (+ 1.0 (* (sqrt (pow (/ 1.0 (- 1.0 (/ hi lo))) 2.0)) (/ (- hi 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 + (sqrt(pow((1.0 / (1.0 - (hi / lo))), 2.0)) * ((hi - 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 + (sqrt(((1.0d0 / (1.0d0 - (hi / lo))) ** 2.0d0)) * ((hi - 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 + (Math.sqrt(Math.pow((1.0 / (1.0 - (hi / lo))), 2.0)) * ((hi - x) / lo));
}

Python

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

↓

def code(lo, hi, x):
	return 1.0 + (math.sqrt(math.pow((1.0 / (1.0 - (hi / lo))), 2.0)) * ((hi - 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(sqrt((Float64(1.0 / Float64(1.0 - Float64(hi / lo))) ^ 2.0)) * Float64(Float64(hi - 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 + (sqrt(((1.0 / (1.0 - (hi / lo))) ^ 2.0)) * ((hi - 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[Sqrt[N[Power[N[(1.0 / N[(1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.87

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

2. Taylor expanded in lo around inf 100

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

3. Simplified 81.14

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

   [Start] 100	\[ \left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo} \]
   +-commutative [=>] 100	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + -1 \cdot \frac{x}{lo}\right)} - -1 \cdot \frac{hi}{lo} \]
   associate--l+ [=>] 100	\[ \color{blue}{\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right)} \]
   +-commutative [=>] 100	\[ \color{blue}{\left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)} + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 100	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\color{blue}{\frac{-1 \cdot x}{lo}} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 100	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\frac{-1 \cdot x}{lo} - \color{blue}{\frac{-1 \cdot hi}{lo}}\right) \]
   div-sub [<=] 100	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{\frac{-1 \cdot x - -1 \cdot hi}{lo}} \]
   distribute-lft-out-- [=>] 100	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \frac{\color{blue}{-1 \cdot \left(x - hi\right)}}{lo} \]
   associate-*r/ [<=] 100	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{-1 \cdot \frac{x - hi}{lo}} \]
   associate-+r+ [<=] 100	\[ \color{blue}{1 + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + -1 \cdot \frac{x - hi}{lo}\right)} \]

4. Applied egg-rr 80.53

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

5. Applied egg-rr 80.53

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

6. Taylor expanded in hi around 0 0.95

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

7. Final simplification 0.95

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

Alternatives

Alternative 1
Error	79.34%
Cost	8896

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

Alternative 2
Error	80.53%
Cost	7616

\[\begin{array}{l} t_0 := 1 + \frac{hi}{lo}\\ 1 + \frac{hi - x}{lo} \cdot \sqrt{t_0 \cdot t_0} \end{array} \]

Alternative 3
Error	80.57%
Cost	6656

\[{\left(\frac{hi}{lo}\right)}^{2} \]

Alternative 4
Error	80.77%
Cost	832

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

Alternative 5
Error	81.21%
Cost	320

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

Alternative 6
Error	81.2%
Cost	256

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

Alternative 7
Error	81.33%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))