2isqrt (example 3.6)

Percentage Accurate: 38.2% → 100.0%

Time: 19.1s

Alternatives: 6

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t\_0} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t\_0}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 0.0)
     (* 0.5 (pow x -1.5))
     (/ (/ 1.0 (+ (sqrt x) t_0)) (sqrt (* x (+ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = (1.0 / (sqrt(x) + t_0)) / sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (1.0d0 / (sqrt(x) + t_0)) / sqrt((x * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / Math.sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / math.sqrt((x * (1.0 + x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / sqrt(Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (1.0 / (sqrt(x) + t_0)) / sqrt((x * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 37.0%

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

   3. Taylor expanded in x around inf 68.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 68.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \]

      2. pow-flip 69.6%

         \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{\color{blue}{{x}^{\left(-3\right)}}}\right) \]

      3. sqrt-pow1 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto 0.5 \cdot \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   7. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 57.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-sub 59.3%

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

      2. *-rgt-identity 59.3%

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

      3. *-un-lft-identity 59.3%

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

      4. +-commutative 59.3%

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

      5. sqrt-unprod 59.3%

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

      6. +-commutative 59.3%

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

   4. Applied egg-rr 59.3%

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

      1. flip-- 84.9%

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

      2. div-inv 84.9%

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

      3. add-sqr-sqrt 82.8%

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

      4. add-sqr-sqrt 99.2%

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

      5. associate--l+ 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-inverses 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. +-commutative 99.2%

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

   8. Simplified 99.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathsf{fma}\left(-0.75, t\_0 \cdot {x}^{-2}, \frac{t\_0}{x}\right) + 0.53125 \cdot \left(t\_0 \cdot {x}^{-3}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (/ 1.0 x)))))
   (+
    (fma -0.75 (* t_0 (pow x -2.0)) (/ t_0 x))
    (* 0.53125 (* t_0 (pow x -3.0))))))

C

double code(double x) {
	double t_0 = 0.5 * sqrt((1.0 / x));
	return fma(-0.75, (t_0 * pow(x, -2.0)), (t_0 / x)) + (0.53125 * (t_0 * pow(x, -3.0)));
}

Julia

function code(x)
	t_0 = Float64(0.5 * sqrt(Float64(1.0 / x)))
	return Float64(fma(-0.75, Float64(t_0 * (x ^ -2.0)), Float64(t_0 / x)) + Float64(0.53125 * Float64(t_0 * (x ^ -3.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.75 * N[(t$95$0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(0.53125 * N[(t$95$0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 38.1%

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

   2. inv-pow 38.1%

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

   3. sqrt-pow2 29.2%

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

   4. metadata-eval 29.2%

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

   5. inv-pow 29.2%

      \[\leadsto e^{\log \left({x}^{-0.5} - \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}\right)} \]

   6. sqrt-pow2 38.3%

      \[\leadsto e^{\log \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right)} \]

   7. +-commutative 38.3%

      \[\leadsto e^{\log \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right)} \]

   8. metadata-eval 38.3%

      \[\leadsto e^{\log \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right)} \]

4. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{e^{\log \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right)}} \]

5. Taylor expanded in x around inf 93.1%

   \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{x}\right) + \left(0.16666666666666666 \cdot \frac{\left(-6 \cdot \left(1 + 0.5 \cdot x\right) + 6 \cdot \left(1 + 0.25 \cdot x\right)\right) - 2}{{x}^{3}} + \left(0.5 \cdot \frac{2 \cdot \left(1 + 0.25 \cdot x\right) - 1}{{x}^{2}} + 2 \cdot \log \left(\frac{1}{x}\right)\right)\right)\right) - \frac{1}{x}}} \]

6. Taylor expanded in x around inf 93.3%

   \[\leadsto \color{blue}{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)} + \left(-0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x} + 0.53125 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{{x}^{2}}\right)} \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.75, \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-2}, \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right) + 0.53125 \cdot \left(\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-3}\right)} \]

9. Final simplification 99.0%

   \[\leadsto \mathsf{fma}\left(-0.75, \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-2}, \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right) + 0.53125 \cdot \left(\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-3}\right) \]
10. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathsf{fma}\left(-0.75, t\_0 \cdot {x}^{-2}, \frac{t\_0}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (/ 1.0 x)))))
   (fma -0.75 (* t_0 (pow x -2.0)) (/ t_0 x))))

C

double code(double x) {
	double t_0 = 0.5 * sqrt((1.0 / x));
	return fma(-0.75, (t_0 * pow(x, -2.0)), (t_0 / x));
}

Julia

function code(x)
	t_0 = Float64(0.5 * sqrt(Float64(1.0 / x)))
	return fma(-0.75, Float64(t_0 * (x ^ -2.0)), Float64(t_0 / x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(-0.75 * N[(t$95$0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 38.1%

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

   2. inv-pow 38.1%

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

   3. sqrt-pow2 29.2%

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

   4. metadata-eval 29.2%

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

   5. inv-pow 29.2%

      \[\leadsto e^{\log \left({x}^{-0.5} - \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}\right)} \]

   6. sqrt-pow2 38.3%

      \[\leadsto e^{\log \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}\right)} \]

   7. +-commutative 38.3%

      \[\leadsto e^{\log \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}\right)} \]

   8. metadata-eval 38.3%

      \[\leadsto e^{\log \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right)} \]

4. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{e^{\log \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right)}} \]

5. Taylor expanded in x around inf 93.1%

   \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{x}\right) + \left(0.16666666666666666 \cdot \frac{\left(-6 \cdot \left(1 + 0.5 \cdot x\right) + 6 \cdot \left(1 + 0.25 \cdot x\right)\right) - 2}{{x}^{3}} + \left(0.5 \cdot \frac{2 \cdot \left(1 + 0.25 \cdot x\right) - 1}{{x}^{2}} + 2 \cdot \log \left(\frac{1}{x}\right)\right)\right)\right) - \frac{1}{x}}} \]

6. Taylor expanded in x around inf 93.2%

   \[\leadsto \color{blue}{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)} + -0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x}} \]
7. Step-by-step derivation

   1. +-commutative 93.2%

      \[\leadsto \color{blue}{-0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x} + e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}} \]

   2. fma-define 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.75, \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x}, e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}\right)} \]

8. Simplified 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.75, \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-2}, \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right)} \]

9. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(-0.75, \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-2}, \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right) \]
10. Add Preprocessing

Alternative 4: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{0.5 \cdot \left(t\_0 + \frac{t\_0 \cdot 0.25 - t\_0}{x}\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (/ (* 0.5 (+ t_0 (/ (- (* t_0 0.25) t_0) x))) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return (0.5 * (t_0 + (((t_0 * 0.25) - t_0) / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 / x))
    code = (0.5d0 * (t_0 + (((t_0 * 0.25d0) - t_0) / x))) / x
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	return (0.5 * (t_0 + (((t_0 * 0.25) - t_0) / x))) / x;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	return (0.5 * (t_0 + (((t_0 * 0.25) - t_0) / x))) / x

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(0.5 * Float64(t_0 + Float64(Float64(Float64(t_0 * 0.25) - t_0) / x))) / x)
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = (0.5 * (t_0 + (((t_0 * 0.25) - t_0) / x))) / x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.5 * N[(t$95$0 + N[(N[(N[(t$95$0 * 0.25), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 38.1%

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

   2. inv-pow 38.1%

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

   3. sqrt-pow2 29.6%

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

   4. metadata-eval 29.6%

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

4. Applied egg-rr 29.6%

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

   1. *-lft-identity 29.6%

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

6. Simplified 29.6%

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

7. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(1 + -0.25 \cdot \frac{x}{{\left(\sqrt{-1}\right)}^{2}}\right)\right) - -0.5 \cdot \sqrt{x}}{x} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]

9. Simplified 85.7%

   \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({x}^{-0.5} \cdot \mathsf{fma}\left(-0.25, \frac{x}{-1}, 1\right) - \sqrt{x}\right)}{x} + \sqrt{x} \cdot 0.5}{{x}^{2}}} \]

10. Taylor expanded in x around inf 98.9%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \frac{0.25 \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}}}{x}}{x}} \]
11. Step-by-step derivation

    1. distribute-lft-out 98.9%

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

    2. *-commutative 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

    \[\leadsto \frac{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \frac{\sqrt{\frac{1}{x}} \cdot 0.25 - \sqrt{\frac{1}{x}}}{x}\right)}{x} \]
14. Add Preprocessing

Alternative 5: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in x around inf 68.3%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 68.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \]

   2. pow-flip 69.2%

      \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{\color{blue}{{x}^{\left(-3\right)}}}\right) \]

   3. sqrt-pow1 97.9%

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

   4. metadata-eval 97.9%

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

   5. metadata-eval 97.9%

      \[\leadsto 0.5 \cdot \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \]

5. Applied egg-rr 97.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \]
6. Step-by-step derivation

   1. *-lft-identity 97.9%

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

7. Simplified 97.9%

   \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

8. Final simplification 97.9%

   \[\leadsto 0.5 \cdot {x}^{-1.5} \]
9. Add Preprocessing

Alternative 6: 5.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -0.5))

C

double code(double x) {
	return pow(x, -0.5);
}

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 38.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 38.1%

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

   2. inv-pow 38.1%

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

   3. sqrt-pow2 29.6%

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

   4. metadata-eval 29.6%

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

4. Applied egg-rr 29.6%

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

   1. *-lft-identity 29.6%

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

6. Simplified 29.6%

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

7. Taylor expanded in x around 0 5.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
8. Step-by-step derivation

   1. unpow1/2 5.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \]

   2. exp-to-pow 5.7%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{x}\right) \cdot 0.5}} \]

   3. log-rec 5.7%

      \[\leadsto e^{\color{blue}{\left(-\log x\right)} \cdot 0.5} \]

   4. distribute-lft-neg-out 5.7%

      \[\leadsto e^{\color{blue}{-\log x \cdot 0.5}} \]

   5. distribute-rgt-neg-in 5.7%

      \[\leadsto e^{\color{blue}{\log x \cdot \left(-0.5\right)}} \]

   6. metadata-eval 5.7%

      \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} \]

   7. exp-to-pow 5.7%

      \[\leadsto \color{blue}{{x}^{-0.5}} \]

9. Simplified 5.7%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]

10. Final simplification 5.7%

    \[\leadsto {x}^{-0.5} \]
11. Add Preprocessing

Developer target: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))

C

double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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]

Reproduce

herbie shell --seed 2024055 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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