xlohi (overflows)

Average Accuracy: 3.1% → 99.1%

Time: 13.5s

Precision: binary64

Cost: 21120

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{t_0}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(1 - \frac{lo}{hi}\right) \cdot {t_0}^{2}} \end{array} \]

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 (* lo (/ (- x lo) (* hi hi))) 2.0)
     (* (- 1.0 (/ lo hi)) (pow t_0 2.0))))))

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((lo * ((x - lo) / (hi * hi))), 2.0) + ((1.0 - (lo / hi)) * pow(t_0, 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
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = (t_0 ** 3.0d0) / (((lo * ((x - lo) / (hi * hi))) ** 2.0d0) + ((1.0d0 - (lo / hi)) * (t_0 ** 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) {
	double t_0 = (x - lo) / hi;
	return Math.pow(t_0, 3.0) / (Math.pow((lo * ((x - lo) / (hi * hi))), 2.0) + ((1.0 - (lo / hi)) * Math.pow(t_0, 2.0)));
}

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((lo * ((x - lo) / (hi * hi))), 2.0) + ((1.0 - (lo / hi)) * math.pow(t_0, 2.0)))

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(lo * Float64(Float64(x - lo) / Float64(hi * hi))) ^ 2.0) + Float64(Float64(1.0 - Float64(lo / hi)) * (t_0 ^ 2.0))))
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) / (((lo * ((x - lo) / (hi * hi))) ^ 2.0) + ((1.0 - (lo / hi)) * (t_0 ^ 2.0)));
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[(lo * N[(N[(x - lo), $MachinePrecision] / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $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 9.5%

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 9.5	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.5	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 9.5%

   \[\leadsto \color{blue}{\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot \frac{1}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)}} \]
   Proof

   [Start] 9.5	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi} \]
   flip3-+ [=>] 9.5	\[ \color{blue}{\frac{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)}} \]
   div-inv [=>] 9.5	\[ \color{blue}{\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot \frac{1}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)}} \]
   pow2 [=>] 9.5	\[ \left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}} + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   pow2 [=>] 9.5	\[ \left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot \frac{1}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left(\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}}} \]
   Proof

   [Start] 9.5	\[ \left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot \frac{1}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-*r/ [=>] 9.5	\[ \color{blue}{\frac{\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}\right) \cdot 1}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)}} \]
   *-rgt-identity [=>] 9.5	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-*r/ [=>] 9.0	\[ \frac{{\color{blue}{\left(\frac{\frac{x - lo}{hi} \cdot lo}{hi}\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-*l/ [<=] 9.5	\[ \frac{{\color{blue}{\left(\frac{\frac{x - lo}{hi}}{hi} \cdot lo\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   *-commutative [<=] 9.5	\[ \frac{{\color{blue}{\left(lo \cdot \frac{\frac{x - lo}{hi}}{hi}\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-/r* [<=] 19.8	\[ \frac{{\left(lo \cdot \color{blue}{\frac{x - lo}{hi \cdot hi}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-*r/ [=>] 15.5	\[ \frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\color{blue}{\left(\frac{\frac{x - lo}{hi} \cdot lo}{hi}\right)}}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-*l/ [<=] 19.8	\[ \frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\color{blue}{\left(\frac{\frac{x - lo}{hi}}{hi} \cdot lo\right)}}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   *-commutative [<=] 19.8	\[ \frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\color{blue}{\left(lo \cdot \frac{\frac{x - lo}{hi}}{hi}\right)}}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-/r* [<=] 99.2	\[ \frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(lo \cdot \color{blue}{\frac{x - lo}{hi \cdot hi}}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]

6. Taylor expanded in hi around inf 0.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(x - lo\right)}^{3}}{{hi}^{3}}}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}} \]

7. Simplified 99.1%

   \[\leadsto \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{3}}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}} \]
   Proof

   [Start] 0.0	\[ \frac{\frac{{\left(x - lo\right)}^{3}}{{hi}^{3}}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}} \]
   cube-div [<=] 99.1	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{3}}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} + 1\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}} \]

8. Final simplification 99.1%

   \[\leadsto \frac{{\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(1 - \frac{lo}{hi}\right) \cdot {\left(\frac{x - lo}{hi}\right)}^{2}} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	14656

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

Alternative 2
Accuracy	99.2%
Cost	14656

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

Alternative 3
Accuracy	19.3%
Cost	6848

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

Alternative 4
Accuracy	18.8%
Cost	320

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

Alternative 5
Accuracy	18.8%
Cost	256

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

Alternative 6
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023143 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))