xlohi (overflows)

Average Accuracy: 3.1% → 98.3%

Time: 11.2s

Precision: binary64

Cost: 21248

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (pow (/ lo hi) 3.0)))
   (-
    (/ x hi)
    (/
     t_0
     (+
      (pow (* (- x lo) (/ lo (* hi hi))) 2.0)
      (- (* (/ lo hi) (/ lo hi)) t_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 = pow((lo / hi), 3.0);
	return (x / hi) - (t_0 / (pow(((x - lo) * (lo / (hi * hi))), 2.0) + (((lo / hi) * (lo / hi)) - t_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 = (lo / hi) ** 3.0d0
    code = (x / hi) - (t_0 / ((((x - lo) * (lo / (hi * hi))) ** 2.0d0) + (((lo / hi) * (lo / hi)) - t_0)))
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((lo / hi), 3.0);
	return (x / hi) - (t_0 / (Math.pow(((x - lo) * (lo / (hi * hi))), 2.0) + (((lo / hi) * (lo / hi)) - t_0)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(lo, hi, x)
	t_0 = (lo / hi) ^ 3.0;
	tmp = (x / hi) - (t_0 / ((((x - lo) * (lo / (hi * hi))) ^ 2.0) + (((lo / hi) * (lo / hi)) - t_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[Power[N[(lo / hi), $MachinePrecision], 3.0], $MachinePrecision]}, N[(N[(x / hi), $MachinePrecision] - N[(t$95$0 / N[(N[Power[N[(N[(x - lo), $MachinePrecision] * N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(lo / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision] - t$95$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 18.8%

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

   [Start] 0.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\frac{x}{hi} + \left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} - \frac{lo}{hi}\right)} \]
   associate-/l* [=>] 18.8	\[ \frac{x}{hi} + \left(\color{blue}{\frac{lo}{\frac{{hi}^{2}}{x - lo}}} - \frac{lo}{hi}\right) \]
   unpow2 [=>] 18.8	\[ \frac{x}{hi} + \left(\frac{lo}{\frac{\color{blue}{hi \cdot hi}}{x - lo}} - \frac{lo}{hi}\right) \]

4. Applied egg-rr 18.8%

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

5. Simplified 64.7%

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

   [Start] 18.8	\[ \frac{x}{hi} + \frac{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{-lo}{hi}\right)}^{3}}{\left({\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \frac{-lo}{hi} \cdot \frac{-lo}{hi}\right) - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}} \]
   associate-*r/ [=>] 0.0	\[ \frac{x}{hi} + \frac{{\color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{hi \cdot hi}\right)}}^{3} + {\left(\frac{-lo}{hi}\right)}^{3}}{\left({\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \frac{-lo}{hi} \cdot \frac{-lo}{hi}\right) - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}} \]
   *-commutative [=>] 0.0	\[ \frac{x}{hi} + \frac{{\left(\frac{\color{blue}{\left(x - lo\right) \cdot lo}}{hi \cdot hi}\right)}^{3} + {\left(\frac{-lo}{hi}\right)}^{3}}{\left({\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \frac{-lo}{hi} \cdot \frac{-lo}{hi}\right) - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}} \]
   associate-*r/ [<=] 18.8	\[ \frac{x}{hi} + \frac{{\color{blue}{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}}^{3} + {\left(\frac{-lo}{hi}\right)}^{3}}{\left({\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \frac{-lo}{hi} \cdot \frac{-lo}{hi}\right) - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}} \]
   distribute-frac-neg [=>] 18.8	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\color{blue}{\left(-\frac{lo}{hi}\right)}}^{3}}{\left({\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \frac{-lo}{hi} \cdot \frac{-lo}{hi}\right) - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}} \]
   associate--l+ [=>] 18.8	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(-\frac{lo}{hi}\right)}^{3}}{\color{blue}{{\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} \cdot \frac{-lo}{hi} - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}\right)}} \]
   associate-*r/ [=>] 0.0	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(-\frac{lo}{hi}\right)}^{3}}{{\color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{hi \cdot hi}\right)}}^{2} + \left(\frac{-lo}{hi} \cdot \frac{-lo}{hi} - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}\right)} \]
   *-commutative [=>] 0.0	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(-\frac{lo}{hi}\right)}^{3}}{{\left(\frac{\color{blue}{\left(x - lo\right) \cdot lo}}{hi \cdot hi}\right)}^{2} + \left(\frac{-lo}{hi} \cdot \frac{-lo}{hi} - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}\right)} \]
   associate-*r/ [<=] 18.8	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(-\frac{lo}{hi}\right)}^{3}}{{\color{blue}{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}}^{2} + \left(\frac{-lo}{hi} \cdot \frac{-lo}{hi} - \left(lo \cdot \frac{x - lo}{hi \cdot hi}\right) \cdot \frac{-lo}{hi}\right)} \]

6. Taylor expanded in x around 0 0.0%

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

7. Simplified 98.3%

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

   [Start] 0.0	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(-\frac{lo}{hi}\right)}^{3}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left(\left(-\frac{lo}{hi}\right) \cdot \left(-\frac{lo}{hi}\right) - \frac{{lo}^{3}}{{hi}^{3}}\right)} \]
   cube-div [<=] 98.3	\[ \frac{x}{hi} + \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(-\frac{lo}{hi}\right)}^{3}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left(\left(-\frac{lo}{hi}\right) \cdot \left(-\frac{lo}{hi}\right) - \color{blue}{{\left(\frac{lo}{hi}\right)}^{3}}\right)} \]

8. Taylor expanded in hi around inf 0.0%

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

9. Simplified 98.3%

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

   [Start] 0.0	\[ \frac{x}{hi} + \frac{-1 \cdot \frac{{lo}^{3}}{{hi}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left(\left(-\frac{lo}{hi}\right) \cdot \left(-\frac{lo}{hi}\right) - {\left(\frac{lo}{hi}\right)}^{3}\right)} \]
   cube-div [<=] 98.3	\[ \frac{x}{hi} + \frac{-1 \cdot \color{blue}{{\left(\frac{lo}{hi}\right)}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left(\left(-\frac{lo}{hi}\right) \cdot \left(-\frac{lo}{hi}\right) - {\left(\frac{lo}{hi}\right)}^{3}\right)} \]
   mul-1-neg [=>] 98.3	\[ \frac{x}{hi} + \frac{\color{blue}{-{\left(\frac{lo}{hi}\right)}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left(\left(-\frac{lo}{hi}\right) \cdot \left(-\frac{lo}{hi}\right) - {\left(\frac{lo}{hi}\right)}^{3}\right)} \]

10. Final simplification 98.3%

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

Alternatives

Alternative 1
Accuracy	19.3%
Cost	7296

\[{\left(1 + \frac{lo}{hi} \cdot 0.3333333333333333\right)}^{3} \cdot \frac{x - lo}{hi} \]

Alternative 2
Accuracy	18.8%
Cost	320

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

Alternative 3
Accuracy	18.8%
Cost	256

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

Alternative 4
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023122 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))