xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%
Time: 19.1s
Alternatives: 6
Speedup: 18.0×

Specification

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

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

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

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

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

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

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

\begin{array}{l}

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

Sampling outcomes in binary64 precision:

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.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
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.

Initial Program: 3.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

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

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

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

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

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

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

\begin{array}{l}

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

Alternative 1: 19.5% accurate, 0.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{lo}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x - hi\right)}\\ 1 - {\left(t\_0 \cdot t\_0\right)}^{-0.5} \end{array} \end{array} \]
(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ lo (fma hi (/ (- x hi) lo) (- x hi)))))
   (- 1.0 (pow (* t_0 t_0) -0.5))))
double code(double lo, double hi, double x) {
	double t_0 = lo / fma(hi, ((x - hi) / lo), (x - hi));
	return 1.0 - pow((t_0 * t_0), -0.5);
}

function code(lo, hi, x)
	t_0 = Float64(lo / fma(hi, Float64(Float64(x - hi) / lo), Float64(x - hi)))
	return Float64(1.0 - (Float64(t_0 * t_0) ^ -0.5))
end

code[lo_, hi_, x_] := Block[{t$95$0 = N[(lo / N[(hi * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] + N[(x - hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{lo}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x - hi\right)}\\
1 - {\left(t\_0 \cdot t\_0\right)}^{-0.5}
\end{array}
\end{array}
Derivation

  1. Initial program 3.1%

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

  3. Taylor expanded in lo around -inf

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

    1. mul-1-negN/A

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

    2. unsub-negN/A

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

    3. lower--.f64N/A

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

    4. lower-/.f64N/A

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

    5. +-commutativeN/A

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

    6. associate--l+N/A

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

    7. associate-/l*N/A

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

    8. lower-fma.f64N/A

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

    9. lower-/.f64N/A

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

    10. lower--.f64N/A

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

    11. lower--.f6418.9

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

  5. Applied rewrites18.9%

    \[\leadsto \color{blue}{1 - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x - hi\right)}{lo}} \]
  6. Step-by-step derivation

    1. Applied rewrites18.9%

      \[\leadsto 1 - \frac{1}{\color{blue}{\frac{lo}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x - hi\right)}}} \]
    2. Step-by-step derivation

      1. Applied rewrites19.7%

        \[\leadsto 1 - {\left(\frac{lo}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x - hi\right)} \cdot \frac{lo}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x - hi\right)}\right)}^{\color{blue}{-0.5}} \]
      2. Add Preprocessing

      Alternative 2: 18.9% accurate, 0.5× speedup?

      \[\begin{array}{l} \\ \mathsf{fma}\left(1 + \frac{hi}{lo}, \frac{hi - x}{lo}, 1\right) \end{array} \]
      (FPCore (lo hi x)
 :precision binary64
 (fma (+ 1.0 (/ hi lo)) (/ (- hi x) lo) 1.0))
      double code(double lo, double hi, double x) {
	return fma((1.0 + (hi / lo)), ((hi - x) / lo), 1.0);
}

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

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

      \begin{array}{l}

\\
\mathsf{fma}\left(1 + \frac{hi}{lo}, \frac{hi - x}{lo}, 1\right)
\end{array}
      Derivation

      1. Initial program 3.1%

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

      3. Taylor expanded in lo around inf

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

      5. Applied rewrites18.9%

        \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{hi}{lo}, \frac{hi - x}{lo}, 1\right)} \]
      6. Add Preprocessing

      Reproduce

      ?
      herbie shell --seed 2024223 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))