xlohi (overflows)

?

Percentage Accurate: 3.1% → 18.8%
Time: 7.6s
Precision: binary64
Cost: 576

?

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} \\ \frac{\frac{lo}{hi} \cdot x - lo}{hi} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x) :precision binary64 (/ (- (* (/ lo hi) x) lo) hi))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return (((lo / hi) * x) - lo) / hi;
}

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 / hi) * x) - lo) / hi
end function

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 / hi) * x) - lo) / hi;
}

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

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

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

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

\frac{x - lo}{hi - lo}
↓
\begin{array}{l}

\\
\frac{\frac{lo}{hi} \cdot x - lo}{hi}
\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 3 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 3.1%

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

  2. Taylor expanded in lo around 0 18.8%

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

  3. Simplified18.8%

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

    [Start]18.8%

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

    mul-1-neg [=>]18.8%

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

    unsub-neg [=>]18.8%

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

    mul-1-neg [=>]18.8%

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

    unsub-neg [=>]18.8%

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

    unpow2 [=>]18.8%

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

  4. Taylor expanded in lo around -inf 18.8%

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

  5. Simplified18.8%

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

    [Start]18.8%

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

    mul-1-neg [=>]18.8%

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

    *-commutative [=>]18.8%

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

    unpow-1 [<=]18.8%

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

    sub-neg [=>]18.8%

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

    mul-1-neg [<=]18.8%

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

    distribute-rgt-in [=>]18.8%

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

    unpow-1 [=>]18.8%

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

    associate-*l/ [=>]18.8%

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

    *-lft-identity [=>]18.8%

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

    associate-*r/ [=>]18.8%

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

    associate-*l/ [=>]10.0%

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

    associate-*r* [<=]10.0%

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

    *-commutative [<=]10.0%

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

    mul-1-neg [=>]10.0%

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

    distribute-neg-frac [<=]10.0%

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

    unsub-neg [=>]10.0%

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

    unpow2 [=>]10.0%

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

    times-frac [=>]18.8%

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

  6. Applied egg-rr18.8%

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

    [Start]18.8%

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

    associate-*r/ [=>]18.8%

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

    sub-div [=>]18.8%

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

  7. Final simplification18.8%

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

Alternatives

Alternative 1
Accuracy18.8%
Cost576
\[\frac{\frac{lo}{hi} \cdot x - lo}{hi} \]
Alternative 2
Accuracy18.8%
Cost256
\[\frac{-lo}{hi} \]
Alternative 3
Accuracy18.7%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023167 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))