xlohi (overflows)

Average Error: 62.0 → 0.5

Time: 12.5s

Precision: binary64

Cost: 21760

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{t_0}^{3}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({t_0}^{2} - \frac{lo}{hi} \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/
    (pow t_0 3.0)
    (+
     (pow (* (- x lo) (/ lo (* hi hi))) 2.0)
     (- (pow t_0 2.0) (* (/ lo hi) (* t_0 t_0)))))))

C

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

↓

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return pow(t_0, 3.0) / (pow(((x - lo) * (lo / (hi * hi))), 2.0) + (pow(t_0, 2.0) - ((lo / hi) * (t_0 * t_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
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = (t_0 ** 3.0d0) / ((((x - lo) * (lo / (hi * hi))) ** 2.0d0) + ((t_0 ** 2.0d0) - ((lo / hi) * (t_0 * t_0))))
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) {
	double t_0 = (x - lo) / hi;
	return Math.pow(t_0, 3.0) / (Math.pow(((x - lo) * (lo / (hi * hi))), 2.0) + (Math.pow(t_0, 2.0) - ((lo / hi) * (t_0 * t_0))));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return math.pow(t_0, 3.0) / (math.pow(((x - lo) * (lo / (hi * hi))), 2.0) + (math.pow(t_0, 2.0) - ((lo / hi) * (t_0 * t_0))))

Julia

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

↓

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return Float64((t_0 ^ 3.0) / Float64((Float64(Float64(x - lo) * Float64(lo / Float64(hi * hi))) ^ 2.0) + Float64((t_0 ^ 2.0) - Float64(Float64(lo / hi) * Float64(t_0 * t_0)))))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = (t_0 ^ 3.0) / ((((x - lo) * (lo / (hi * hi))) ^ 2.0) + ((t_0 ^ 2.0) - ((lo / hi) * (t_0 * t_0))));
end

Wolfram

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

↓

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[(N[Power[N[(N[(x - lo), $MachinePrecision] * N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(N[(lo / hi), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0

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

2. Taylor expanded in hi around inf 64.0

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

3. Simplified 57.9

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

   [Start] 64.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 64.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   *-commutative [=>] 64.0	\[ \frac{\color{blue}{\left(x - lo\right) \cdot lo}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   unpow2 [=>] 64.0	\[ \frac{\left(x - lo\right) \cdot lo}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 57.9	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 57.9	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 0.5

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

5. Taylor expanded in hi around inf 64.0

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

6. Simplified 0.5

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

   [Start] 64.0	\[ \frac{\frac{{\left(x - lo\right)}^{3}}{{hi}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   cube-div [<=] 0.5	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]

7. Applied egg-rr 0.5

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

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.6
Cost	21760

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{t_0}^{3}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left(\frac{t_0}{\frac{hi}{x - lo}} - {t_0}^{2} \cdot \frac{lo}{hi}\right)} \end{array} \]

Alternative 2
Error	0.5
Cost	14656

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {t_0}^{2}}{t_0 \cdot \left(\frac{lo}{hi} + -1\right)} \end{array} \]

Alternative 3
Error	50.8
Cost	6848

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

Alternative 4
Error	51.6
Cost	448

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

Alternative 5
Error	52.0
Cost	256

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

Alternative 6
Error	52.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023056 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))