2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.5%
Time: 10.8s
Alternatives: 8
Speedup: 1.8×

Specification

?
\[x > 1 \land x < 10^{+308}\]
\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
\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 8 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: 38.7% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
\end{array}

Alternative 1: 98.5% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \frac{-0.5}{x} \cdot \frac{\frac{1 - x}{\sqrt{x}} - \frac{\mathsf{fma}\left(0.25, x, 1\right)}{\sqrt{x} \cdot x}}{x} \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (/ -0.5 x)
  (/ (- (/ (- 1.0 x) (sqrt x)) (/ (fma 0.25 x 1.0) (* (sqrt x) x))) x)))
double code(double x) {
	return (-0.5 / x) * ((((1.0 - x) / sqrt(x)) - (fma(0.25, x, 1.0) / (sqrt(x) * x))) / x);
}

function code(x)
	return Float64(Float64(-0.5 / x) * Float64(Float64(Float64(Float64(1.0 - x) / sqrt(x)) - Float64(fma(0.25, x, 1.0) / Float64(sqrt(x) * x))) / x))
end

code[x_] := N[(N[(-0.5 / x), $MachinePrecision] * N[(N[(N[(N[(1.0 - x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(0.25 * x + 1.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{-0.5}{x} \cdot \frac{\frac{1 - x}{\sqrt{x}} - \frac{\mathsf{fma}\left(0.25, x, 1\right)}{\sqrt{x} \cdot x}}{x}
\end{array}
Derivation

  1. Initial program 38.5%

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
  2. Add Preprocessing

  3. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

  5. Applied rewrites81.3%

    \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

  7. Applied rewrites98.9%

    \[\leadsto \frac{\frac{1 - x}{\sqrt{x}} - \frac{\mathsf{fma}\left(0.25, x, 1\right)}{\sqrt{x} \cdot x}}{x} \cdot \color{blue}{\frac{-0.5}{x}} \]

  8. Final simplification98.9%

    \[\leadsto \frac{-0.5}{x} \cdot \frac{\frac{1 - x}{\sqrt{x}} - \frac{\mathsf{fma}\left(0.25, x, 1\right)}{\sqrt{x} \cdot x}}{x} \]
  9. Add Preprocessing

Alternative 2: 97.6% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \end{array} \]
(FPCore (x)
 :precision binary64
 (/ (fma -0.125 (sqrt (/ 1.0 (* (* x x) x))) (* (sqrt (/ 1.0 x)) 0.5)) x))
double code(double x) {
	return fma(-0.125, sqrt((1.0 / ((x * x) * x))), (sqrt((1.0 / x)) * 0.5)) / x;
}

function code(x)
	return Float64(fma(-0.125, sqrt(Float64(1.0 / Float64(Float64(x * x) * x))), Float64(sqrt(Float64(1.0 / x)) * 0.5)) / x)
end

code[x_] := N[(N[(-0.125 * N[Sqrt[N[(1.0 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

\begin{array}{l}

\\
\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}
\end{array}
Derivation

  1. Initial program 38.7%

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
  2. Add Preprocessing

  4. Applied rewrites40.8%

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

  5. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x}} \]
  6. Step-by-step derivation

    1. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x}} \]

    2. lower-fma.f64N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}}{x} \]

    3. lower-sqrt.f64N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    4. lower-/.f64N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    5. unpow3N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot x}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    6. unpow2N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{{x}^{2}} \cdot x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    7. lower-*.f64N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{{x}^{2} \cdot x}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    8. unpow2N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    9. lower-*.f64N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    10. lower-*.f64N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}}\right)}{x} \]

    11. lower-sqrt.f64N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)}{x} \]

    12. lower-/.f6497.6

      \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right)}{x} \]

  7. Applied rewrites97.6%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]

  8. Final simplification97.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]
  9. Add Preprocessing

Developer Target 1: 98.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]
(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))
double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

public static double code(double x) {
	return 1.0 / (((x + 1.0) * Math.sqrt(x)) + (x * Math.sqrt((x + 1.0))));
}

def code(x):
	return 1.0 / (((x + 1.0) * math.sqrt(x)) + (x * math.sqrt((x + 1.0))))

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

function tmp = code(x)
	tmp = 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
end

code[x_] := N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}
\end{array}

Developer Target 2: 38.7% accurate, 0.2× speedup?

\[\begin{array}{l} \\ {x}^{-0.5} - {\left(x + 1\right)}^{-0.5} \end{array} \]
(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))
double code(double x) {
	return pow(x, -0.5) - pow((x + 1.0), -0.5);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
end function

public static double code(double x) {
	return Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
}

def code(x):
	return math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)

function code(x)
	return Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5))
end

function tmp = code(x)
	tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
end

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}
\end{array}

Reproduce

?
herbie shell --seed 2024229 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))