xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 22.3s

Alternatives: 6

Speedup: 7.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{hi}{lo}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ hi lo) 2.0))

C

double code(double lo, double hi, double x) {
	return pow((hi / lo), 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 = (hi / lo) ** 2.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow((hi / lo), 2.0);
}

Python

def code(lo, hi, x):
	return math.pow((hi / lo), 2.0)

Julia

function code(lo, hi, x)
	return Float64(hi / lo) ^ 2.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi / lo) ^ 2.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in x around 0 9.9%

   \[\leadsto -1 \cdot \frac{x - lo}{lo} + \color{blue}{hi \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
5. Step-by-step derivation

   1. distribute-rgt-in 9.9%

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

   2. associate-*l/ 9.9%

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

   3. *-lft-identity 9.9%

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

   4. associate-*l/ 0.0%

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

   5. unpow2 0.0%

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

   6. times-frac 18.9%

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

   7. unpow2 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around inf 0.0%

   \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \left(\frac{hi}{lo \cdot x} + \frac{{hi}^{2}}{{lo}^{2} \cdot x}\right)\right) - \frac{1}{lo}\right)} \]
8. Step-by-step derivation

   1. sub-neg 0.0%

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

   2. distribute-neg-frac 0.0%

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

   3. metadata-eval 0.0%

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

   4. +-commutative 0.0%

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

   5. associate-+l+ 0.0%

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

   6. associate-/r* 0.0%

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

   7. associate-/r* 0.0%

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

   8. unpow2 0.0%

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

   9. unpow2 0.0%

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

   10. times-frac 18.9%

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

   11. unpow2 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around inf 0.0%

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

    2. unpow2 0.0%

       \[\leadsto \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}} \]

    3. times-frac 19.5%

       \[\leadsto \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} \]

    4. unpow2 19.5%

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

12. Simplified 19.5%

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

13. Final simplification 19.5%

    \[\leadsto {\left(\frac{hi}{lo}\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return ((lo - x) / lo) + ((hi / lo) + ((hi / lo) * (hi / lo)));
}

Fortran

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) / lo) + ((hi / lo) + ((hi / lo) * (hi / lo)))
end function

Java

public static double code(double lo, double hi, double x) {
	return ((lo - x) / lo) + ((hi / lo) + ((hi / lo) * (hi / lo)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in x around 0 9.9%

   \[\leadsto -1 \cdot \frac{x - lo}{lo} + \color{blue}{hi \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
5. Step-by-step derivation

   1. distribute-rgt-in 9.9%

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

   2. associate-*l/ 9.9%

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

   3. *-lft-identity 9.9%

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

   4. associate-*l/ 0.0%

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

   5. unpow2 0.0%

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

   6. times-frac 18.9%

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

   7. unpow2 18.9%

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

6. Simplified 18.9%

   \[\leadsto -1 \cdot \frac{x - lo}{lo} + \color{blue}{\left(\frac{hi}{lo} + {\left(\frac{hi}{lo}\right)}^{2}\right)} \]
7. Step-by-step derivation

   1. unpow2 18.9%

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

8. Applied egg-rr 18.9%

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

9. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} + \left(\frac{hi}{lo} + \frac{hi}{lo} \cdot \frac{hi}{lo}\right) \]
10. Add Preprocessing

Alternative 3: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (((x - hi) / lo) * (-1.0 - (hi / lo))) + 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 - hi) / lo) * ((-1.0d0) - (hi / lo))) + 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return (((x - hi) / lo) * (-1.0 - (hi / lo))) + 1.0;
}

Python

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

Julia

function code(lo, hi, x)
	return Float64(Float64(Float64(Float64(x - hi) / lo) * Float64(-1.0 - Float64(hi / lo))) + 1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (+ (* hi (/ (+ (/ hi lo) 1.0) lo)) 1.0))

C

double code(double lo, double hi, double x) {
	return (hi * (((hi / lo) + 1.0) / lo)) + 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 = (hi * (((hi / lo) + 1.0d0) / lo)) + 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return (hi * (((hi / lo) + 1.0) / lo)) + 1.0;
}

Python

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

Julia

function code(lo, hi, x)
	return Float64(Float64(hi * Float64(Float64(Float64(hi / lo) + 1.0) / lo)) + 1.0)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi * (((hi / lo) + 1.0) / lo)) + 1.0;
end

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

   \[\leadsto 1 + \color{blue}{\frac{hi \cdot \left(1 + \frac{hi}{lo}\right)}{lo}} \]
7. Step-by-step derivation

   1. associate-/l* 18.9%

      \[\leadsto 1 + \color{blue}{hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]

8. Simplified 18.9%

   \[\leadsto 1 + \color{blue}{hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]

9. Final simplification 18.9%

   \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
10. Add Preprocessing

Alternative 5: 18.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ -\frac{lo}{hi} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return -(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 = -(lo / hi)
end function

Java

public static double code(double lo, double hi, double x) {
	return -(lo / hi);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{lo}{hi}} \]
5. Step-by-step derivation

   1. associate-*r/ 18.8%

      \[\leadsto \color{blue}{\frac{-1 \cdot lo}{hi}} \]

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto -\frac{lo}{hi} \]
8. Add Preprocessing

Alternative 6: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 1.0)

C

double code(double lo, double hi, double x) {
	return 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 = 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0;
}

Python

def code(lo, hi, x):
	return 1.0

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0;
end

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))