xlohi (overflows)

Average Error: 62.0 → 52.0

Time: 9.0s

Precision: binary64

Cost: 704

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

Math

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

↓

\[\frac{x - lo}{hi} \cdot 2 + \frac{lo}{hi} \]

FPCore

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

↓

(FPCore (lo hi x) :precision binary64 (+ (* (/ (- x lo) hi) 2.0) (/ lo hi)))

C

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

↓

double code(double lo, double hi, double x) {
	return (((x - lo) / hi) * 2.0) + (lo / hi);
}

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 = (((x - lo) / hi) * 2.0d0) + (lo / hi)
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 (((x - lo) / hi) * 2.0) + (lo / hi);
}

Python

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

↓

def code(lo, hi, x):
	return (((x - lo) / hi) * 2.0) + (lo / hi)

Julia

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

↓

function code(lo, hi, x)
	return Float64(Float64(Float64(Float64(x - lo) / hi) * 2.0) + Float64(lo / hi))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	tmp = (((x - lo) / hi) * 2.0) + (lo / hi);
end

Wolfram

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

↓

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

Derivation

1. Initial program 62.0

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

2. Taylor expanded in hi around inf 52.0

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

3. Applied egg-rr 52.0

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

4. Applied egg-rr 52.0

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

5. Simplified 52.0

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

   [Start] 52.0	\[ lo \cdot \frac{2}{hi} + \left(\frac{x - lo}{hi} \cdot 2 - \frac{x + lo}{hi}\right) \]
   rational_best-simplify-49 [<=] 52.0	\[ \color{blue}{\frac{x - lo}{hi} \cdot 2 - \left(\frac{x + lo}{hi} - lo \cdot \frac{2}{hi}\right)} \]
   rational_best-simplify-46 [<=] 52.0	\[ \color{blue}{lo \cdot \frac{2}{hi} - \left(\frac{x + lo}{hi} - \frac{x - lo}{hi} \cdot 2\right)} \]
   rational_best-simplify-49 [=>] 52.0	\[ \color{blue}{\frac{x - lo}{hi} \cdot 2 + \left(lo \cdot \frac{2}{hi} - \frac{x + lo}{hi}\right)} \]

6. Taylor expanded in lo around inf 52.0

   \[\leadsto \frac{x - lo}{hi} \cdot 2 + \color{blue}{\frac{lo}{hi}} \]

7. Final simplification 52.0

   \[\leadsto \frac{x - lo}{hi} \cdot 2 + \frac{lo}{hi} \]

Alternatives

Alternative 1
Error	52.0
Cost	320

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

Alternative 2
Error	52.0
Cost	256

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

Alternative 3
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023096 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))