xlohi (overflows)

Average Accuracy: 3.1% → 98.6%

Time: 11.7s

Precision: binary64

Cost: 34944

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{hi}{x - lo}\right)}^{-2}\\ t_1 := \frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)\\ \frac{{t_1}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left({t_1}^{2} + t_0\right) - \frac{t_0}{\frac{hi}{lo}}} \end{array} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (/ hi (- x lo)) -2.0)) (t_1 (* (/ lo (* hi hi)) (- x lo))))
   (/
    (+ (pow t_1 3.0) (pow (/ (- x lo) hi) 3.0))
    (- (+ (pow t_1 2.0) t_0) (/ t_0 (/ hi lo))))))

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 = pow((hi / (x - lo)), -2.0);
	double t_1 = (lo / (hi * hi)) * (x - lo);
	return (pow(t_1, 3.0) + pow(((x - lo) / hi), 3.0)) / ((pow(t_1, 2.0) + t_0) - (t_0 / (hi / lo)));
}

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
    real(8) :: t_1
    t_0 = (hi / (x - lo)) ** (-2.0d0)
    t_1 = (lo / (hi * hi)) * (x - lo)
    code = ((t_1 ** 3.0d0) + (((x - lo) / hi) ** 3.0d0)) / (((t_1 ** 2.0d0) + t_0) - (t_0 / (hi / lo)))
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 = Math.pow((hi / (x - lo)), -2.0);
	double t_1 = (lo / (hi * hi)) * (x - lo);
	return (Math.pow(t_1, 3.0) + Math.pow(((x - lo) / hi), 3.0)) / ((Math.pow(t_1, 2.0) + t_0) - (t_0 / (hi / lo)));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = math.pow((hi / (x - lo)), -2.0)
	t_1 = (lo / (hi * hi)) * (x - lo)
	return (math.pow(t_1, 3.0) + math.pow(((x - lo) / hi), 3.0)) / ((math.pow(t_1, 2.0) + t_0) - (t_0 / (hi / lo)))

Julia

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

↓

function code(lo, hi, x)
	t_0 = Float64(hi / Float64(x - lo)) ^ -2.0
	t_1 = Float64(Float64(lo / Float64(hi * hi)) * Float64(x - lo))
	return Float64(Float64((t_1 ^ 3.0) + (Float64(Float64(x - lo) / hi) ^ 3.0)) / Float64(Float64((t_1 ^ 2.0) + t_0) - Float64(t_0 / Float64(hi / lo))))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	t_0 = (hi / (x - lo)) ^ -2.0;
	t_1 = (lo / (hi * hi)) * (x - lo);
	tmp = ((t_1 ^ 3.0) + (((x - lo) / hi) ^ 3.0)) / (((t_1 ^ 2.0) + t_0) - (t_0 / (hi / lo)));
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[Power[N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision] * N[(x - lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + N[Power[N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision] - N[(t$95$0 / N[(hi / lo), $MachinePrecision]), $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 9.6%

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 9.6	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.6	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Taylor expanded in hi around 0 0.0%

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

5. Simplified 9.6%

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

   [Start] 0.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   unpow2 [=>] 0.0	\[ \left(\frac{lo \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}} + \frac{x}{hi}\right) - \frac{lo}{hi} \]
   times-frac [=>] 9.6	\[ \left(\color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}} + \frac{x}{hi}\right) - \frac{lo}{hi} \]
   *-commutative [<=] 9.6	\[ \left(\color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \frac{x}{hi}\right) - \frac{lo}{hi} \]
   associate-+r- [<=] 9.6	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   div-sub [<=] 9.6	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]
   *-rgt-identity [<=] 9.6	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi} \cdot 1} \]
   distribute-lft-in [<=] 9.6	\[ \color{blue}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right)} \]

6. Applied egg-rr 9.9%

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

   [Start] 9.6	\[ \frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + 1\right) \]
   distribute-lft-in [=>] 9.6	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi} \cdot 1} \]
   flip3-+ [=>] 9.6	\[ \color{blue}{\frac{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi} \cdot 1\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\left(\frac{x - lo}{hi} \cdot 1\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right) - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right)\right)}} \]
   clear-num [=>] 9.6	\[ \frac{{\left(\frac{x - lo}{hi} \cdot \color{blue}{\frac{1}{\frac{hi}{lo}}}\right)}^{3} + {\left(\frac{x - lo}{hi} \cdot 1\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\left(\frac{x - lo}{hi} \cdot 1\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right) - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right)\right)} \]
   frac-times [=>] 10.3	\[ \frac{{\color{blue}{\left(\frac{\left(x - lo\right) \cdot 1}{hi \cdot \frac{hi}{lo}}\right)}}^{3} + {\left(\frac{x - lo}{hi} \cdot 1\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\left(\frac{x - lo}{hi} \cdot 1\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right) - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right)\right)} \]
   *-commutative [<=] 10.3	\[ \frac{{\left(\frac{\color{blue}{1 \cdot \left(x - lo\right)}}{hi \cdot \frac{hi}{lo}}\right)}^{3} + {\left(\frac{x - lo}{hi} \cdot 1\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\left(\frac{x - lo}{hi} \cdot 1\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right) - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right)\right)} \]
   *-un-lft-identity [<=] 10.3	\[ \frac{{\left(\frac{\color{blue}{x - lo}}{hi \cdot \frac{hi}{lo}}\right)}^{3} + {\left(\frac{x - lo}{hi} \cdot 1\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\left(\frac{x - lo}{hi} \cdot 1\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right) - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right)\right)} \]
   *-rgt-identity [=>] 10.3	\[ \frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{3} + {\color{blue}{\left(\frac{x - lo}{hi}\right)}}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\left(\frac{x - lo}{hi} \cdot 1\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right) - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot 1\right)\right)} \]

7. Simplified 98.6%

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

   [Start] 9.9	\[ \frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   associate-*r/ [=>] 19.4	\[ \frac{{\left(\frac{x - lo}{\color{blue}{\frac{hi \cdot hi}{lo}}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   unpow2 [<=] 19.4	\[ \frac{{\left(\frac{x - lo}{\frac{\color{blue}{{hi}^{2}}}{lo}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   associate-/l* [<=] 0.0	\[ \frac{{\color{blue}{\left(\frac{\left(x - lo\right) \cdot lo}{{hi}^{2}}\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   *-commutative [<=] 0.0	\[ \frac{{\left(\frac{\color{blue}{lo \cdot \left(x - lo\right)}}{{hi}^{2}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   associate-/l* [=>] 19.4	\[ \frac{{\color{blue}{\left(\frac{lo}{\frac{{hi}^{2}}{x - lo}}\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   associate-/r/ [=>] 19.4	\[ \frac{{\color{blue}{\left(\frac{lo}{{hi}^{2}} \cdot \left(x - lo\right)\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   unpow2 [=>] 19.4	\[ \frac{{\left(\frac{lo}{\color{blue}{hi \cdot hi}} \cdot \left(x - lo\right)\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + \left({\left(\frac{hi}{x - lo}\right)}^{-2} - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}\right)} \]
   associate-+r- [=>] 19.4	\[ \frac{{\left(\frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\color{blue}{\left({\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} + {\left(\frac{hi}{x - lo}\right)}^{-2}\right) - \frac{x - lo}{hi \cdot \frac{hi}{lo}} \cdot \frac{x - lo}{hi}}} \]

8. Final simplification 98.6%

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

Alternatives

Alternative 1
Accuracy	98.5%
Cost	704

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

Alternative 2
Accuracy	98.5%
Cost	704

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

Alternative 3
Accuracy	98.5%
Cost	448

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

Alternative 4
Accuracy	18.8%
Cost	256

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

Alternative 5
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))