xlohi (overflows)

?

Percentage Accurate: 3.1% → 99.2%
Time: 11.6s
Precision: binary64
Cost: 1472

?

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \frac{lo - x}{hi}\\ \frac{t_0 \cdot t_0}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- lo x) hi)))
   (/ (* t_0 t_0) (* (/ (- x lo) hi) (- 1.0 (/ lo hi))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	double t_0 = (lo - x) / hi;
	return (t_0 * t_0) / (((x - lo) / hi) * (1.0 - (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
    real(8) :: t_0
    t_0 = (lo - x) / hi
    code = (t_0 * t_0) / (((x - lo) / hi) * (1.0d0 - (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) {
	double t_0 = (lo - x) / hi;
	return (t_0 * t_0) / (((x - lo) / hi) * (1.0 - (lo / hi)));
}

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

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	t_0 = Float64(Float64(lo - x) / hi)
	return Float64(Float64(t_0 * t_0) / Float64(Float64(Float64(x - lo) / hi) * Float64(1.0 - Float64(lo / hi))))
end

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end
↓
function tmp = code(lo, hi, x)
	t_0 = (lo - x) / hi;
	tmp = (t_0 * t_0) / (((x - lo) / hi) * (1.0 - (lo / hi)));
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(lo - x), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] * N[(1.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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

Local Percentage Accuracy?

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.

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Results

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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. Simplified9.1%

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

    [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 [=>]9.1

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

    div-sub [<=]9.1

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

  4. Applied egg-rr9.1%

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

    [Start]9.1

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

    flip-+ [=>]9.1

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

    frac-2neg [=>]9.1

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

    pow2 [=>]9.1

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

    pow2 [=>]9.1

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

    *-commutative [=>]9.1

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

    *-un-lft-identity [=>]9.1

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

    distribute-rgt-out-- [=>]9.1

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

  5. Simplified99.3%

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

    [Start]9.1

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

    distribute-rgt-neg-in [=>]9.1

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

    neg-sub0 [=>]9.1

    \[ \frac{\color{blue}{0 - \left({\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}\right)}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]

    associate--r- [=>]9.1

    \[ \frac{\color{blue}{\left(0 - {\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{2}\right) + {\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(-\left(\frac{lo}{hi} - 1\right)\right)} \]

    neg-sub0 [<=]9.1

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

    +-commutative [<=]9.1

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

    sub-neg [<=]9.1

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

    associate-*r/ [=>]8.9

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

    associate-*l/ [<=]9.1

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

    *-commutative [<=]9.1

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

    associate-/r* [<=]99.3

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

    sub-neg [=>]99.3

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

    metadata-eval [=>]99.3

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

  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. Simplified99.3%

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

    [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 [=>]99.3

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

    unpow2 [<=]99.3

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

  8. Applied egg-rr0.0%

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

    [Start]99.3

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

    unpow2 [=>]99.3

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

    frac-2neg [=>]99.3

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

    frac-2neg [=>]99.3

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

    frac-times [=>]0.0

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

    sub-neg [=>]0.0

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

    distribute-neg-in [=>]0.0

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

    remove-double-neg [=>]0.0

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

    sub-neg [=>]0.0

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

    distribute-neg-in [=>]0.0

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

    remove-double-neg [=>]0.0

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

  9. Simplified99.3%

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

    [Start]0.0

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

    sqr-neg [=>]0.0

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

    times-frac [=>]99.3

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

    neg-mul-1 [=>]99.3

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

    +-commutative [=>]99.3

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

    neg-mul-1 [<=]99.3

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

    unsub-neg [=>]99.3

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

    neg-mul-1 [=>]99.3

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

    +-commutative [=>]99.3

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

    neg-mul-1 [<=]99.3

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

    unsub-neg [=>]99.3

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

  10. Final simplification99.3%

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

Alternatives

Alternative 1
Accuracy98.4%
Cost640
\[\frac{\frac{-lo}{hi}}{1 - \frac{lo}{hi}} \]
Alternative 2
Accuracy19.2%
Cost448
\[\frac{lo}{hi} \cdot \frac{lo}{hi} \]
Alternative 3
Accuracy18.8%
Cost320
\[\frac{x - lo}{hi} \]
Alternative 4
Accuracy18.8%
Cost256
\[\frac{-lo}{hi} \]
Alternative 5
Accuracy18.7%
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 2023160 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))