xlohi (overflows)

Average Error: 62.0 → 48.6

Time: 10.0s

Precision: binary64

Cost: 1088

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

Math

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

↓

\[\left(\left(x + x\right) - lo\right) \cdot \left(2 \cdot \frac{0.25}{hi}\right) + \frac{lo}{hi \cdot -2} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (+ (* (- (+ x x) lo) (* 2.0 (/ 0.25 hi))) (/ lo (* hi -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 (((x + x) - lo) * (2.0 * (0.25 / hi))) + (lo / (hi * -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 = (((x + x) - lo) * (2.0d0 * (0.25d0 / hi))) + (lo / (hi * (-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 (((x + x) - lo) * (2.0 * (0.25 / hi))) + (lo / (hi * -2.0));
}

Python

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

↓

def code(lo, hi, x):
	return (((x + x) - lo) * (2.0 * (0.25 / hi))) + (lo / (hi * -2.0))

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 + x) - lo) * Float64(2.0 * Float64(0.25 / hi))) + Float64(lo / Float64(hi * -2.0)))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	tmp = (((x + x) - lo) * (2.0 * (0.25 / hi))) + (lo / (hi * -2.0));
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 + x), $MachinePrecision] - lo), $MachinePrecision] * N[(2.0 * N[(0.25 / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(lo / N[(hi * -2.0), $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

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

4. Applied egg-rr 48.5

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

5. Applied egg-rr 64.0

   \[\leadsto \color{blue}{\left(2 \cdot \left(\left(x + x\right) - lo\right)\right) \cdot \frac{0.25}{hi}} + \frac{lo}{hi \cdot -2} \]

6. Simplified 48.6

   \[\leadsto \color{blue}{\left(\left(x + x\right) - lo\right) \cdot \left(2 \cdot \frac{0.25}{hi}\right)} + \frac{lo}{hi \cdot -2} \]
   Proof

   [Start] 64.0	\[ \left(2 \cdot \left(\left(x + x\right) - lo\right)\right) \cdot \frac{0.25}{hi} + \frac{lo}{hi \cdot -2} \]
   rational_best-simplify-1 [<=] 64.0	\[ \color{blue}{\frac{0.25}{hi} \cdot \left(2 \cdot \left(\left(x + x\right) - lo\right)\right)} + \frac{lo}{hi \cdot -2} \]
   rational_best-simplify-50 [=>] 48.6	\[ \color{blue}{\left(\left(x + x\right) - lo\right) \cdot \left(2 \cdot \frac{0.25}{hi}\right)} + \frac{lo}{hi \cdot -2} \]

7. Final simplification 48.6

   \[\leadsto \left(\left(x + x\right) - lo\right) \cdot \left(2 \cdot \frac{0.25}{hi}\right) + \frac{lo}{hi \cdot -2} \]

Alternatives

Alternative 1
Error	48.5
Cost	832

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

Alternative 2
Error	48.5
Cost	576

\[-\left(\frac{-lo}{lo + lo} - 0.5\right) \]

Alternative 3
Error	48.5
Cost	384

\[-\left(\frac{x}{lo} - 0.5\right) \]

Alternative 4
Error	52.0
Cost	320

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

Alternative 5
Error	52.0
Cost	256

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

Alternative 6
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023100 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))