xlohi (overflows)

Average Error: 62.0 → 0.3

Time: 11.6s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x) :precision binary64 (/ (/ (- lo x) hi) (+ (/ 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) {
	return ((lo - x) / hi) / ((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
    code = ((lo - x) / hi) / ((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) {
	return ((lo - x) / hi) / ((lo / hi) + -1.0);
}

Python

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

↓

def code(lo, hi, x):
	return ((lo - x) / hi) / ((lo / hi) + -1.0)

Julia

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

↓

function code(lo, hi, x)
	return Float64(Float64(Float64(lo - x) / hi) / 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)
	tmp = ((lo - x) / hi) / ((lo / hi) + -1.0);
end

Wolfram

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

↓

code[lo_, hi_, x_] := N[(N[(N[(lo - x), $MachinePrecision] / hi), $MachinePrecision] / N[(N[(lo / hi), $MachinePrecision] + -1.0), $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 58.1

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 58.1	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 58.1	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

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 0.6

   \[\leadsto \color{blue}{\left(0 - e^{\mathsf{log1p}\left(\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}\right)}\right) + 1} \]

8. Simplified 0.3

   \[\leadsto \color{blue}{\frac{-\frac{x - lo}{hi}}{\frac{lo}{hi} + -1}} \]
   Proof

   [Start] 0.6	\[ \left(0 - e^{\mathsf{log1p}\left(\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}\right)}\right) + 1 \]
   associate-+l- [=>] 0.6	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}\right)} - 1\right)} \]
   expm1-def [=>] 0.5	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}\right)\right)} \]
   expm1-log1p [=>] 0.5	\[ 0 - \color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}} \]
   sub0-neg [=>] 0.5	\[ \color{blue}{-\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}} \]
   distribute-neg-frac [=>] 0.5	\[ \color{blue}{\frac{-{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}} \]
   mul-1-neg [<=] 0.5	\[ \frac{\color{blue}{-1 \cdot {\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)} \]
   associate-/l* [=>] 0.5	\[ \color{blue}{\frac{-1}{\frac{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} + -1\right)}{{\left(\frac{x - lo}{hi}\right)}^{2}}}} \]
   *-commutative [=>] 0.5	\[ \frac{-1}{\frac{\color{blue}{\left(\frac{lo}{hi} + -1\right) \cdot \frac{x - lo}{hi}}}{{\left(\frac{x - lo}{hi}\right)}^{2}}} \]
   associate-/l* [=>] 0.4	\[ \frac{-1}{\color{blue}{\frac{\frac{lo}{hi} + -1}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi}}}}} \]
   associate-/l* [<=] 18.5	\[ \frac{-1}{\frac{\frac{lo}{hi} + -1}{\color{blue}{\frac{{\left(\frac{x - lo}{hi}\right)}^{2} \cdot hi}{x - lo}}}} \]
   associate-*r/ [<=] 0.6	\[ \frac{-1}{\frac{\frac{lo}{hi} + -1}{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2} \cdot \frac{hi}{x - lo}}}} \]

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	51.5
Cost	576

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

Alternative 2
Error	51.5
Cost	448

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

Alternative 3
Error	52.0
Cost	256

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

Alternative 4
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023067 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))