xlohi (overflows)

?

Percentage Accurate: 3.1% → 99.3%
Time: 13.0s
Precision: binary64
Cost: 14144

?

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{-{t_0}^{2}}{\mathsf{fma}\left(\frac{lo}{hi}, t_0, \frac{lo - x}{hi}\right)} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/ (- (pow t_0 2.0)) (fma (/ lo hi) t_0 (/ (- lo x) 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 = (x - lo) / hi;
	return -pow(t_0, 2.0) / fma((lo / hi), t_0, ((lo - x) / hi));
}

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 ^ 2.0)) / fma(Float64(lo / hi), t_0, Float64(Float64(lo - x) / 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[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[((-N[Power[t$95$0, 2.0], $MachinePrecision]) / N[(N[(lo / hi), $MachinePrecision] * t$95$0 + N[(N[(lo - x), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{x - lo}{hi - lo}
↓
\begin{array}{l}
t_0 := \frac{x - lo}{hi}\\
\frac{-{t_0}^{2}}{\mathsf{fma}\left(\frac{lo}{hi}, t_0, \frac{lo - x}{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.

Herbie found 7 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

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. Simplified9.9%

    \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
    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 [=>]9.9

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

    div-sub [<=]9.9

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

  4. Applied egg-rr9.9%

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

    [Start]9.9

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

    clear-num [=>]9.9

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

    un-div-inv [=>]9.9

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

  5. Applied egg-rr9.9%

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

    [Start]9.9

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

    flip-+ [=>]9.9

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

    div-sub [=>]9.9

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

  6. Simplified99.4%

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

    [Start]9.9

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

    div-sub [<=]9.9

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

    *-commutative [=>]9.9

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

    associate-*l/ [=>]9.6

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

    associate-*r/ [<=]9.9

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

    associate-/r* [<=]99.5

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

    *-commutative [=>]99.5

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

    fma-neg [=>]99.4

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

    distribute-frac-neg [<=]99.4

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

    neg-sub0 [=>]99.4

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

    associate--r- [=>]99.4

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

    neg-sub0 [<=]99.4

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

  7. Taylor expanded in hi around inf 0.0%

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

  8. Simplified99.4%

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

    [Start]0.0

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

    mul-1-neg [=>]0.0

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

    unpow2 [=>]0.0

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

    unpow2 [=>]0.0

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

    times-frac [=>]99.4

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

    unpow2 [<=]99.4

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

  9. Final simplification99.4%

    \[\leadsto \frac{-{\left(\frac{x - lo}{hi}\right)}^{2}}{\mathsf{fma}\left(\frac{lo}{hi}, \frac{x - lo}{hi}, \frac{lo - x}{hi}\right)} \]

Alternatives

Alternative 1
Accuracy98.8%
Cost7552
\[\frac{{\left(\frac{hi}{x - lo}\right)}^{-2}}{\frac{x - lo}{hi} \cdot \left(1 - \frac{lo}{hi}\right)} \]
Alternative 2
Accuracy99.2%
Cost7552
\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{t_0}^{2}}{t_0 \cdot \left(1 - \frac{lo}{hi}\right)} \end{array} \]
Alternative 3
Accuracy20.6%
Cost6720
\[\mathsf{log1p}\left(\frac{x - lo}{hi}\right) \]
Alternative 4
Accuracy18.8%
Cost576
\[\frac{lo \cdot \left(-1 + \frac{x}{hi}\right)}{hi} \]
Alternative 5
Accuracy18.8%
Cost256
\[\frac{-lo}{hi} \]
Alternative 6
Accuracy18.7%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023163 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))