xlohi (overflows)

Average Accuracy: 3.1% → 99.2%

Time: 12.0s

Precision: binary64

Cost: 1472

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi))) (/ (* t_0 t_0) (* 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;
	return (t_0 * t_0) / (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
    t_0 = (x - lo) / hi
    code = (t_0 * t_0) / (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;
	return (t_0 * t_0) / (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
	return (t_0 * t_0) / (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)
	return Float64(Float64(t_0 * t_0) / 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;
	tmp = (t_0 * t_0) / (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]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(t$95$0 * N[(1.0 - N[(lo / hi), $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 9.4%

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 9.4	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.4	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 9.4%

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

   [Start] 9.4	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi} \]
   flip-+ [=>] 9.4	\[ \color{blue}{\frac{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}}} \]
   frac-2neg [=>] 9.4	\[ \color{blue}{\frac{-\left(\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}\right)}{-\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}\right)}} \]
   pow2 [=>] 9.4	\[ \frac{-\left(\color{blue}{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}} - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}\right)}{-\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}\right)} \]
   pow2 [=>] 9.4	\[ \frac{-\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}\right)}{-\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi} - \frac{x - lo}{hi}\right)} \]
   *-commutative [=>] 9.4	\[ \frac{-\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}{-\left(\color{blue}{\frac{lo}{hi} \cdot \frac{x - lo}{hi}} - \frac{x - lo}{hi}\right)} \]
   *-un-lft-identity [=>] 9.4	\[ \frac{-\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}{-\left(\frac{lo}{hi} \cdot \frac{x - lo}{hi} - \color{blue}{1 \cdot \frac{x - lo}{hi}}\right)} \]
   distribute-rgt-out-- [=>] 9.4	\[ \frac{-\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}{-\color{blue}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)}} \]

5. Simplified 99.2%

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

   [Start] 9.4	\[ \frac{-\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}{-\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   distribute-rgt-neg-in [=>] 9.4	\[ \frac{-\left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}{\color{blue}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)}} \]
   neg-sub0 [=>] 9.4	\[ \frac{\color{blue}{0 - \left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   associate--r- [=>] 9.4	\[ \frac{\color{blue}{\left(0 - {\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}\right) + {\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   neg-sub0 [<=] 9.4	\[ \frac{\color{blue}{\left(-{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}\right)} + {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   +-commutative [<=] 9.4	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2} + \left(-{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}\right)}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   sub-neg [<=] 9.4	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   associate-*r/ [=>] 9.0	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{\frac{x - lo}{hi} \cdot lo}{hi}\right)}}^{2}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   associate-*l/ [<=] 9.4	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(\frac{\frac{x - lo}{hi}}{hi} \cdot lo\right)}}^{2}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   *-commutative [<=] 9.4	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\color{blue}{\left(lo \cdot \frac{\frac{x - lo}{hi}}{hi}\right)}}^{2}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   associate-/r* [<=] 99.2	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \color{blue}{\frac{x - lo}{hi \cdot hi}}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]
   sub-neg [=>] 99.2	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(-\color{blue}{\left(\frac{lo}{hi} + \left(-1\right)\right)}\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2} - {\left(lo \cdot \frac{x - lo}{hi \cdot hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} + \color{blue}{-1}\right)\right)} \]

6. Taylor expanded in hi around inf 0.0%

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

7. Simplified 99.2%

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

   [Start] 0.0	\[ \frac{\frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]
   unpow2 [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]
   unpow2 [=>] 0.0	\[ \frac{\frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]
   times-frac [=>] 99.2	\[ \frac{\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]
   unpow2 [<=] 99.2	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]

8. Applied egg-rr 99.2%

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

   [Start] 99.2	\[ \frac{{\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]
   unpow2 [=>] 99.2	\[ \frac{\color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{-lo}{hi} + 1\right)} \]

9. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	98.5%
Cost	640

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

Alternative 2
Accuracy	18.8%
Cost	256

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

Alternative 3
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023136 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))