xlohi (overflows)

Average Accuracy: 3.1% → 99.1%

Time: 13.9s

Precision: binary64

Cost: 8960

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/
    (+ (pow (* x (/ lo (* hi hi))) 2.0) (- (* t_0 (/ lo hi)) (* t_0 (/ x hi))))
    (* t_0 (+ (/ lo hi) -1.0)))))

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((x * (lo / (hi * hi))), 2.0) + ((t_0 * (lo / hi)) - (t_0 * (x / hi)))) / (t_0 * ((lo / hi) + -1.0));
}

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 = (((x * (lo / (hi * hi))) ** 2.0d0) + ((t_0 * (lo / hi)) - (t_0 * (x / hi)))) / (t_0 * ((lo / hi) + (-1.0d0)))
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) {
	double t_0 = (x - lo) / hi;
	return (Math.pow((x * (lo / (hi * hi))), 2.0) + ((t_0 * (lo / hi)) - (t_0 * (x / hi)))) / (t_0 * ((lo / hi) + -1.0));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return (math.pow((x * (lo / (hi * hi))), 2.0) + ((t_0 * (lo / hi)) - (t_0 * (x / hi)))) / (t_0 * ((lo / hi) + -1.0))

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(Float64((Float64(x * Float64(lo / Float64(hi * hi))) ^ 2.0) + Float64(Float64(t_0 * Float64(lo / hi)) - Float64(t_0 * Float64(x / hi)))) / Float64(t_0 * Float64(Float64(lo / hi) + -1.0)))
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 = (((x * (lo / (hi * hi))) ^ 2.0) + ((t_0 * (lo / hi)) - (t_0 * (x / hi)))) / (t_0 * ((lo / hi) + -1.0));
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[(N[Power[N[(x * N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * N[(lo / hi), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(x / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(lo / hi), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

3. Simplified 9.4%

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

   [Start] 0.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   *-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(x - lo\right) \cdot lo}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   unpow2 [=>] 0.0	\[ \frac{\left(x - lo\right) \cdot lo}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 9.4	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.4	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 99.1%

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

   [Start] 9.4	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi} \]
   flip-+ [=>] 9.4	\[ \color{blue}{\frac{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}}} \]
   pow2 [=>] 9.4	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   associate-*l/ [=>] 9.0	\[ \frac{{\color{blue}{\left(\frac{\left(x - lo\right) \cdot \frac{lo}{hi}}{hi}\right)}}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   *-un-lft-identity [=>] 9.0	\[ \frac{{\left(\frac{\left(x - lo\right) \cdot \frac{lo}{hi}}{\color{blue}{1 \cdot hi}}\right)}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   times-frac [=>] 9.4	\[ \frac{{\color{blue}{\left(\frac{x - lo}{1} \cdot \frac{\frac{lo}{hi}}{hi}\right)}}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   flip-- [=>] 0.0	\[ \frac{{\left(\frac{\color{blue}{\frac{x \cdot x - lo \cdot lo}{x + lo}}}{1} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   associate-/l/ [=>] 0.0	\[ \frac{{\left(\color{blue}{\frac{x \cdot x - lo \cdot lo}{1 \cdot \left(x + lo\right)}} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   *-un-lft-identity [<=] 0.0	\[ \frac{{\left(\frac{x \cdot x - lo \cdot lo}{\color{blue}{x + lo}} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   flip-- [<=] 9.4	\[ \frac{{\left(\color{blue}{\left(x - lo\right)} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   associate-/l/ [=>] 99.4	\[ \frac{{\left(\left(x - lo\right) \cdot \color{blue}{\frac{lo}{hi \cdot hi}}\right)}^{2} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   pow2 [=>] 99.4	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}} \]
   *-commutative [=>] 99.4	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}} - \frac{x - lo}{hi}} \]
   *-un-lft-identity [=>] 99.4	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo}{hi} \cdot \frac{x - lo}{hi} - \color{blue}{1 \cdot \frac{x - lo}{hi}}} \]

5. Applied egg-rr 99.1%

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

   [Start] 99.1	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [=>] 99.1	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - \color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   div-sub [=>] 99.1	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - \frac{x - lo}{hi} \cdot \color{blue}{\left(\frac{x}{hi} - \frac{lo}{hi}\right)}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   sub-neg [=>] 99.1	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - \frac{x - lo}{hi} \cdot \color{blue}{\left(\frac{x}{hi} + \left(-\frac{lo}{hi}\right)\right)}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   distribute-lft-in [=>] 99.1	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} - \color{blue}{\left(\frac{x - lo}{hi} \cdot \frac{x}{hi} + \frac{x - lo}{hi} \cdot \left(-\frac{lo}{hi}\right)\right)}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]

6. Taylor expanded in x around inf 49.5%

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

7. Simplified 99.1%

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

   [Start] 49.5	\[ \frac{{\left(\frac{lo \cdot x}{{hi}^{2}}\right)}^{2} - \left(\frac{x - lo}{hi} \cdot \frac{x}{hi} + \frac{x - lo}{hi} \cdot \left(-\frac{lo}{hi}\right)\right)}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   *-commutative [=>] 49.5	\[ \frac{{\left(\frac{\color{blue}{x \cdot lo}}{{hi}^{2}}\right)}^{2} - \left(\frac{x - lo}{hi} \cdot \frac{x}{hi} + \frac{x - lo}{hi} \cdot \left(-\frac{lo}{hi}\right)\right)}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [=>] 49.5	\[ \frac{{\left(\frac{x \cdot lo}{\color{blue}{hi \cdot hi}}\right)}^{2} - \left(\frac{x - lo}{hi} \cdot \frac{x}{hi} + \frac{x - lo}{hi} \cdot \left(-\frac{lo}{hi}\right)\right)}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   associate-*r/ [<=] 99.1	\[ \frac{{\color{blue}{\left(x \cdot \frac{lo}{hi \cdot hi}\right)}}^{2} - \left(\frac{x - lo}{hi} \cdot \frac{x}{hi} + \frac{x - lo}{hi} \cdot \left(-\frac{lo}{hi}\right)\right)}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]

8. Final simplification 99.1%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	7616

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

Alternative 2
Accuracy	19.5%
Cost	7232

\[1 + \left|-1 - \frac{hi}{lo}\right| \cdot \frac{hi - x}{lo} \]

Alternative 3
Accuracy	19.3%
Cost	6720

\[\left|\frac{hi - x}{lo}\right| \]

Alternative 4
Accuracy	19.3%
Cost	6592

\[\left|\frac{hi}{lo}\right| \]

Alternative 5
Accuracy	19.4%
Cost	576

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

Alternative 6
Accuracy	19.5%
Cost	448

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

Alternative 7
Accuracy	18.8%
Cost	256

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

Alternative 8
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))