xlohi (overflows)

Average Error: 62.0 → 52.0

Time: 16.9s

Precision: binary64

Cost: 640

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double lo, double hi, double x) {
	return -(lo / hi) + (2.0 * (x / 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 = -(lo / hi) + (2.0d0 * (x / 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 -(lo / hi) + (2.0 * (x / hi));
}

Python

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

   \[\leadsto \color{blue}{\frac{2}{hi \cdot \frac{\frac{hi}{x - lo}}{x - lo}} \cdot \left(hi \cdot \frac{-0.5}{lo - x}\right)} \]

4. Taylor expanded in lo around inf 52.0

   \[\leadsto \frac{2}{hi \cdot \frac{\frac{hi}{x - lo}}{x - lo}} \cdot \color{blue}{\left(-0.5 \cdot \frac{hi}{lo}\right)} \]

5. Taylor expanded in x around 0 52.0

   \[\leadsto \color{blue}{2 \cdot \frac{x}{hi} + -1 \cdot \frac{lo}{hi}} \]

6. Simplified 52.0

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

   [Start] 52.0	\[ 2 \cdot \frac{x}{hi} + -1 \cdot \frac{lo}{hi} \]
   rational.json-simplify-1 [=>] 52.0	\[ \color{blue}{-1 \cdot \frac{lo}{hi} + 2 \cdot \frac{x}{hi}} \]
   rational.json-simplify-2 [=>] 52.0	\[ \color{blue}{\frac{lo}{hi} \cdot -1} + 2 \cdot \frac{x}{hi} \]
   rational.json-simplify-9 [=>] 52.0	\[ \color{blue}{\left(-\frac{lo}{hi}\right)} + 2 \cdot \frac{x}{hi} \]

7. Final simplification 52.0

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

Alternatives

Alternative 1
Error	52.0
Cost	576

\[4 \cdot \left(\left(x - lo\right) \cdot \frac{0.25}{hi}\right) \]

Alternative 2
Error	52.0
Cost	320

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

Alternative 3
Error	52.0
Cost	256

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

Alternative 4
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))