xlohi (overflows)

Average Error: 62.0 → 0.4

Time: 11.6s

Precision: binary64

Cost: 7872

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- lo x) hi)))
   (- (/ (pow (/ (- x lo) hi) 2.0) (- t_0 (* (/ lo hi) 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 = (lo - x) / hi;
	return -(pow(((x - lo) / hi), 2.0) / (t_0 - ((lo / hi) * 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 = (lo - x) / hi
    code = -((((x - lo) / hi) ** 2.0d0) / (t_0 - ((lo / hi) * 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 = (lo - x) / hi;
	return -(Math.pow(((x - lo) / hi), 2.0) / (t_0 - ((lo / hi) * t_0)));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = (lo - x) / hi
	return -(math.pow(((x - lo) / hi), 2.0) / (t_0 - ((lo / hi) * 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(lo - x) / hi)
	return Float64(-Float64((Float64(Float64(x - lo) / hi) ^ 2.0) / Float64(t_0 - Float64(Float64(lo / hi) * t_0))))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	t_0 = (lo - x) / hi;
	tmp = -((((x - lo) / hi) ^ 2.0) / (t_0 - ((lo / hi) * 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[(lo - x), $MachinePrecision] / hi), $MachinePrecision]}, (-N[(N[Power[N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$0 - N[(N[(lo / hi), $MachinePrecision] * t$95$0), $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 58.0

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 58.0	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 58.0	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 0.5

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

5. Taylor expanded in hi around inf 64.0

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

6. Simplified 0.5

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

   [Start] 64.0	\[ \frac{-1 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   mul-1-neg [=>] 64.0	\[ \frac{\color{blue}{-\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [=>] 64.0	\[ \frac{-\frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [=>] 64.0	\[ \frac{-\frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   times-frac [=>] 0.5	\[ \frac{-\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [<=] 0.5	\[ \frac{-\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]

7. Applied egg-rr 0.4

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

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	51.9
Cost	704

\[1 + \frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right) \]

Alternative 2
Error	0.4
Cost	704

\[\frac{\frac{lo - x}{hi}}{\frac{lo}{hi} + -1} \]

Alternative 3
Error	52.0
Cost	320

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

Alternative 4
Error	52.0
Cost	256

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

Alternative 5
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)))