xlohi (overflows)

Average Error: 62.0 → 0.5

Time: 11.9s

Precision: binary64

Cost: 2752

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := \frac{lo}{hi \cdot hi}\\ \frac{\left(t_0 + \left(x - lo\right) \cdot t_1\right) \cdot \left(t_0 + t_1 \cdot \left(lo - x\right)\right)}{t_0 \cdot \left(1 - \frac{lo}{hi}\right)} \end{array} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)) (t_1 (/ lo (* hi hi))))
   (/
    (* (+ t_0 (* (- x lo) t_1)) (+ t_0 (* t_1 (- lo x))))
    (* t_0 (- 1.0 (/ lo hi))))))

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;
	double t_1 = lo / (hi * hi);
	return ((t_0 + ((x - lo) * t_1)) * (t_0 + (t_1 * (lo - x)))) / (t_0 * (1.0 - (lo / hi)));
}

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 = (x - lo) / hi
    t_1 = lo / (hi * hi)
    code = ((t_0 + ((x - lo) * t_1)) * (t_0 + (t_1 * (lo - x)))) / (t_0 * (1.0d0 - (lo / hi)))
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;
	double t_1 = lo / (hi * hi);
	return ((t_0 + ((x - lo) * t_1)) * (t_0 + (t_1 * (lo - x)))) / (t_0 * (1.0 - (lo / hi)));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	t_1 = lo / (hi * hi)
	return ((t_0 + ((x - lo) * t_1)) * (t_0 + (t_1 * (lo - x)))) / (t_0 * (1.0 - (lo / hi)))

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)
	t_1 = Float64(lo / Float64(hi * hi))
	return Float64(Float64(Float64(t_0 + Float64(Float64(x - lo) * t_1)) * Float64(t_0 + Float64(t_1 * Float64(lo - x)))) / Float64(t_0 * Float64(1.0 - Float64(lo / hi))))
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;
	t_1 = lo / (hi * hi);
	tmp = ((t_0 + ((x - lo) * t_1)) * (t_0 + (t_1 * (lo - x)))) / (t_0 * (1.0 - (lo / hi)));
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]}, Block[{t$95$1 = N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 + N[(N[(x - lo), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[(t$95$1 * N[(lo - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $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 57.9

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 57.9	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 57.9	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 57.9

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

5. Applied egg-rr 57.9

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

6. Simplified 0.5

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

   [Start] 57.9	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{\frac{x - lo}{hi}}{\frac{hi}{lo}}\right)}^{2}}{\frac{x - lo}{hi} - \frac{\frac{x - lo}{hi}}{\frac{hi}{lo}}} \]
   associate-/r/ [=>] 57.9	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{\frac{x - lo}{hi}}{hi} \cdot lo\right)}}^{2}}{\frac{x - lo}{hi} - \frac{\frac{x - lo}{hi}}{\frac{hi}{lo}}} \]
   associate-/r* [<=] 0.5	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\color{blue}{\frac{x - lo}{hi \cdot hi}} \cdot lo\right)}^{2}}{\frac{x - lo}{hi} - \frac{\frac{x - lo}{hi}}{\frac{hi}{lo}}} \]
   *-commutative [<=] 0.5	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}}^{2}}{\frac{x - lo}{hi} - \frac{\frac{x - lo}{hi}}{\frac{hi}{lo}}} \]
   *-lft-identity [<=] 0.5	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{\color{blue}{1 \cdot \frac{x - lo}{hi}} - \frac{\frac{x - lo}{hi}}{\frac{hi}{lo}}} \]
   associate-/l* [<=] 17.2	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{1 \cdot \frac{x - lo}{hi} - \color{blue}{\frac{\frac{x - lo}{hi} \cdot lo}{hi}}} \]
   associate-*r/ [<=] 0.4	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{1 \cdot \frac{x - lo}{hi} - \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}}} \]
   *-commutative [<=] 0.4	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{1 \cdot \frac{x - lo}{hi} - \color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}}} \]
   distribute-rgt-out-- [=>] 0.5	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{\color{blue}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)}} \]

7. Applied egg-rr 0.5

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

8. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\left(\frac{x - lo}{hi} + lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \left(\frac{x - lo}{hi} - lo \cdot \frac{x - lo}{hi \cdot hi}\right)}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   associate-*r/ [=>] 64.0	\[ \frac{\left(\frac{x - lo}{hi} + \color{blue}{\frac{lo \cdot \left(x - lo\right)}{hi \cdot hi}}\right) \cdot \left(\frac{x - lo}{hi} - lo \cdot \frac{x - lo}{hi \cdot hi}\right)}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   associate-*l/ [<=] 0.5	\[ \frac{\left(\frac{x - lo}{hi} + \color{blue}{\frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)}\right) \cdot \left(\frac{x - lo}{hi} - lo \cdot \frac{x - lo}{hi \cdot hi}\right)}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   associate-*r/ [=>] 64.0	\[ \frac{\left(\frac{x - lo}{hi} + \frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)\right) \cdot \left(\frac{x - lo}{hi} - \color{blue}{\frac{lo \cdot \left(x - lo\right)}{hi \cdot hi}}\right)}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
   associate-*l/ [<=] 0.5	\[ \frac{\left(\frac{x - lo}{hi} + \frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)\right) \cdot \left(\frac{x - lo}{hi} - \color{blue}{\frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)}\right)}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]

9. Final simplification 0.5

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

Alternatives

Alternative 1
Error	51.6
Cost	832

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

Alternative 2
Error	51.5
Cost	448

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

Alternative 3
Error	52.0
Cost	320

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

Alternative 4
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023011 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))