2isqrt (example 3.6)

?

Percentage Accurate: 69.1% → 99.7%
Time: 10.7s
Precision: binary64
Cost: 27332

?

\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{x + 1}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \end{array} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sqrt (+ x 1.0)))))
   (if (<= (+ (/ 1.0 (sqrt x)) t_0) 1e-8)
     (*
      (pow x -0.5)
      (+ (/ 0.5 x) (- (/ 0.3125 (pow x 3.0)) (/ 0.375 (* x x)))))
     (+ (pow x -0.5) t_0))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	double t_0 = -1.0 / sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + t_0) <= 1e-8) {
		tmp = pow(x, -0.5) * ((0.5 / x) + ((0.3125 / pow(x, 3.0)) - (0.375 / (x * x))));
	} else {
		tmp = pow(x, -0.5) + t_0;
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + t_0) <= 1d-8) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) - (0.375d0 / (x * x))))
    else
        tmp = (x ** (-0.5d0)) + t_0
    end if
    code = tmp
end function

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}
↓
public static double code(double x) {
	double t_0 = -1.0 / Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + t_0) <= 1e-8) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) - (0.375 / (x * x))));
	} else {
		tmp = Math.pow(x, -0.5) + t_0;
	}
	return tmp;
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))
↓
def code(x):
	t_0 = -1.0 / math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + t_0) <= 1e-8:
		tmp = math.pow(x, -0.5) * ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) - (0.375 / (x * x))))
	else:
		tmp = math.pow(x, -0.5) + t_0
	return tmp

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	t_0 = Float64(-1.0 / sqrt(Float64(x + 1.0)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + t_0) <= 1e-8)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) - Float64(0.375 / Float64(x * x)))));
	else
		tmp = Float64((x ^ -0.5) + t_0);
	end
	return tmp
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp_2 = code(x)
	t_0 = -1.0 / sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + t_0) <= 1e-8)
		tmp = (x ^ -0.5) * ((0.5 / x) + ((0.3125 / (x ^ 3.0)) - (0.375 / (x * x))));
	else
		tmp = (x ^ -0.5) + t_0;
	end
	tmp_2 = tmp;
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]
↓
code[x_] := Block[{t$95$0 = N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], 1e-8], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + t$95$0), $MachinePrecision]]]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{-1}{\sqrt{x + 1}}\\
\mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 10^{-8}:\\
\;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.5} + t_0\\


\end{array}
\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 13 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

Target

Original69.1%
Target99.1%
Herbie99.7%
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation?

  1. Split input into 2 regimes

  2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1e-8

    1. Initial program 42.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

    2. Applied egg-rr42.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
      Step-by-step derivation

      [Start]42.0%

      \[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

      frac-sub [=>]42.0%

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

      div-inv [=>]42.0%

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

      *-un-lft-identity [<=]42.0%

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

      +-commutative [=>]42.0%

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

      *-rgt-identity [=>]42.0%

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

      metadata-eval [<=]42.0%

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

      frac-times [<=]42.0%

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

      un-div-inv [=>]42.0%

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

      pow1/2 [=>]42.0%

      \[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]

      pow-flip [=>]42.0%

      \[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]

      metadata-eval [=>]42.0%

      \[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]

      +-commutative [=>]42.0%

      \[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

    3. Simplified42.0%

      \[\leadsto \color{blue}{\left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot {x}^{-0.5}} \]
      Step-by-step derivation

      [Start]42.0%

      \[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]

      associate-*r/ [=>]42.0%

      \[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]

      *-rgt-identity [<=]42.0%

      \[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot 1}} \]

      times-frac [=>]42.0%

      \[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      div-sub [=>]42.0%

      \[ \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{1 + x}} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right)} \cdot \frac{{x}^{-0.5}}{1} \]

      *-inverses [=>]42.0%

      \[ \left(\color{blue}{1} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \frac{{x}^{-0.5}}{1} \]

      /-rgt-identity [=>]42.0%

      \[ \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \color{blue}{{x}^{-0.5}} \]

    4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]

    5. Simplified99.7%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)} \cdot {x}^{-0.5} \]
      Step-by-step derivation

      [Start]99.7%

      \[ \left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      associate--l+ [=>]99.7%

      \[ \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {x}^{-0.5} \]

      associate-*r/ [=>]99.7%

      \[ \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      metadata-eval [=>]99.7%

      \[ \left(\frac{\color{blue}{0.5}}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      associate-*r/ [=>]99.7%

      \[ \left(\frac{0.5}{x} + \left(\color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      metadata-eval [=>]99.7%

      \[ \left(\frac{0.5}{x} + \left(\frac{\color{blue}{0.3125}}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      associate-*r/ [=>]99.7%

      \[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right)\right) \cdot {x}^{-0.5} \]

      metadata-eval [=>]99.7%

      \[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{\color{blue}{0.375}}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      unpow2 [=>]99.7%

      \[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{\color{blue}{x \cdot x}}\right)\right) \cdot {x}^{-0.5} \]

    if 1e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

    1. Initial program 99.4%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

    2. Applied egg-rr99.8%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{\sqrt{x + 1}} \]
      Step-by-step derivation

      [Start]99.4%

      \[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

      add-log-exp [=>]6.3%

      \[ \color{blue}{\log \left(e^{\frac{1}{\sqrt{x}}}\right)} - \frac{1}{\sqrt{x + 1}} \]

      *-un-lft-identity [=>]6.3%

      \[ \log \color{blue}{\left(1 \cdot e^{\frac{1}{\sqrt{x}}}\right)} - \frac{1}{\sqrt{x + 1}} \]

      log-prod [=>]6.3%

      \[ \color{blue}{\left(\log 1 + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

      metadata-eval [=>]6.3%

      \[ \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) - \frac{1}{\sqrt{x + 1}} \]

      add-log-exp [<=]99.4%

      \[ \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) - \frac{1}{\sqrt{x + 1}} \]

      pow1/2 [=>]99.4%

      \[ \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - \frac{1}{\sqrt{x + 1}} \]

      pow-flip [=>]99.8%

      \[ \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - \frac{1}{\sqrt{x + 1}} \]

      metadata-eval [=>]99.8%

      \[ \left(0 + {x}^{\color{blue}{-0.5}}\right) - \frac{1}{\sqrt{x + 1}} \]

    3. Simplified99.8%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
      Step-by-step derivation

      [Start]99.8%

      \[ \left(0 + {x}^{-0.5}\right) - \frac{1}{\sqrt{x + 1}} \]

      +-lft-identity [=>]99.8%

      \[ \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification99.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.7%
Cost27332
\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{x + 1}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]
Alternative 2
Accuracy99.4%
Cost26756
\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{x + 1}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]
Alternative 3
Accuracy99.3%
Cost13888
\[\frac{-1}{x} \cdot \frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(-1 - x\right)} \]
Alternative 4
Accuracy99.7%
Cost13380
\[\begin{array}{l} \mathbf{if}\;x \leq 175000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]
Alternative 5
Accuracy99.0%
Cost7300
\[\begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]
Alternative 6
Accuracy98.6%
Cost7172
\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]
Alternative 7
Accuracy98.5%
Cost7044
\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]
Alternative 8
Accuracy98.3%
Cost6916
\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]
Alternative 9
Accuracy66.9%
Cost6852
\[\begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]
Alternative 10
Accuracy49.4%
Cost6656
\[-1 + {x}^{-0.5} \]
Alternative 11
Accuracy2.2%
Cost6592
\[-{x}^{-0.5} \]
Alternative 12
Accuracy49.5%
Cost6528
\[{x}^{-0.5} \]
Alternative 13
Accuracy2.0%
Cost64
\[-1 \]

Reproduce?

herbie shell --seed 2023167 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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