2isqrt (example 3.6)

Percentage Accurate: 39.7% → 98.7%

Time: 9.6s

Alternatives: 8

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: 39.7% 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: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|{x}^{-2.5}\right|\\ \frac{\left(\mathsf{fma}\left(t\_0, -0.25, {x}^{-0.5} \cdot 0.5\right) - {x}^{-0.5} \cdot \frac{0.375}{x}\right) + t\_0 \cdot 0.5}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (pow x -2.5))))
   (/
    (+
     (- (fma t_0 -0.25 (* (pow x -0.5) 0.5)) (* (pow x -0.5) (/ 0.375 x)))
     (* t_0 0.5))
    x)))

C

double code(double x) {
	double t_0 = fabs(pow(x, -2.5));
	return ((fma(t_0, -0.25, (pow(x, -0.5) * 0.5)) - (pow(x, -0.5) * (0.375 / x))) + (t_0 * 0.5)) / x;
}

Julia

function code(x)
	t_0 = abs((x ^ -2.5))
	return Float64(Float64(Float64(fma(t_0, -0.25, Float64((x ^ -0.5) * 0.5)) - Float64((x ^ -0.5) * Float64(0.375 / x))) + Float64(t_0 * 0.5)) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Power[x, -2.5], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(t$95$0 * -0.25 + N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf 81.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(-0.25 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) - -0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}}{x}} \]

6. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\left|{x}^{-2.5}\right|, -0.25, {x}^{-0.5} \cdot 0.5\right) - {x}^{-0.5} \cdot \frac{0.375}{x}\right) + \left|{x}^{-2.5}\right| \cdot 0.5}{x}} \]
7. Add Preprocessing

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma -1.0 (/ 0.375 (pow x 1.5)) (* 0.5 (sqrt (/ 1.0 x)))) x))

C

double code(double x) {
	return fma(-1.0, (0.375 / pow(x, 1.5)), (0.5 * sqrt((1.0 / x)))) / x;
}

Julia

function code(x)
	return Float64(fma(-1.0, Float64(0.375 / (x ^ 1.5)), Float64(0.5 * sqrt(Float64(1.0 / x)))) / x)
end

Wolfram

code[x_] := N[(N[(-1.0 * N[(0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf 81.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. fma-define 99.0%

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

   2. distribute-rgt-out 99.0%

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

   3. metadata-eval 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   4. *-commutative 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}\right)}{x} \]

6. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]
7. Step-by-step derivation

   1. associate-/l* 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   2. inv-pow 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \sqrt{\color{blue}{{x}^{-1}}} \cdot \frac{0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   3. sqrt-pow1 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \frac{0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   4. metadata-eval 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, {x}^{\color{blue}{-0.5}} \cdot \frac{0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   5. *-un-lft-identity 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{1 \cdot \left({x}^{-0.5} \cdot \frac{0.375}{x}\right)}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   6. metadata-eval 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \left({x}^{\color{blue}{\left(-0.5\right)}} \cdot \frac{0.375}{x}\right), \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   7. pow-flip 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \left(\color{blue}{\frac{1}{{x}^{0.5}}} \cdot \frac{0.375}{x}\right), \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   8. pow1/2 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \left(\frac{1}{\color{blue}{\sqrt{x}}} \cdot \frac{0.375}{x}\right), \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   9. frac-times 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \color{blue}{\frac{1 \cdot 0.375}{\sqrt{x} \cdot x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   10. metadata-eval 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \frac{\color{blue}{0.375}}{\sqrt{x} \cdot x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   11. *-commutative 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \frac{0.375}{\color{blue}{x \cdot \sqrt{x}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   12. pow1 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \frac{0.375}{\color{blue}{{x}^{1}} \cdot \sqrt{x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   13. pow1/2 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \frac{0.375}{{x}^{1} \cdot \color{blue}{{x}^{0.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   14. pow-prod-up 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \frac{0.375}{\color{blue}{{x}^{\left(1 + 0.5\right)}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   15. metadata-eval 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-1, 1 \cdot \frac{0.375}{{x}^{\color{blue}{1.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

8. Applied egg-rr 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{1 \cdot \frac{0.375}{{x}^{1.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]
9. Step-by-step derivation

   1. *-lft-identity 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{\frac{0.375}{{x}^{1.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

10. Simplified 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{\frac{0.375}{{x}^{1.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

11. Final simplification 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]
12. Add Preprocessing

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (* (pow x -0.5) 0.5) (* (pow x -0.5) (/ 0.375 x))) x))

C

double code(double x) {
	return ((pow(x, -0.5) * 0.5) - (pow(x, -0.5) * (0.375 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x ** (-0.5d0)) * 0.5d0) - ((x ** (-0.5d0)) * (0.375d0 / x))) / x
end function

Java

public static double code(double x) {
	return ((Math.pow(x, -0.5) * 0.5) - (Math.pow(x, -0.5) * (0.375 / x))) / x;
}

Python

def code(x):
	return ((math.pow(x, -0.5) * 0.5) - (math.pow(x, -0.5) * (0.375 / x))) / x

Julia

function code(x)
	return Float64(Float64(Float64((x ^ -0.5) * 0.5) - Float64((x ^ -0.5) * Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((x ^ -0.5) * 0.5) - ((x ^ -0.5) * (0.375 / x))) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf 81.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

6. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot 0.5 - {x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]
7. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{\sqrt{x}} + \frac{-0.375}{{x}^{1.5}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ 0.5 (sqrt x)) (/ -0.375 (pow x 1.5))) x))

C

double code(double x) {
	return ((0.5 / sqrt(x)) + (-0.375 / pow(x, 1.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 / sqrt(x)) + ((-0.375d0) / (x ** 1.5d0))) / x
end function

Java

public static double code(double x) {
	return ((0.5 / Math.sqrt(x)) + (-0.375 / Math.pow(x, 1.5))) / x;
}

Python

def code(x):
	return ((0.5 / math.sqrt(x)) + (-0.375 / math.pow(x, 1.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.5 / sqrt(x)) + Float64(-0.375 / (x ^ 1.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 / sqrt(x)) + (-0.375 / (x ^ 1.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf 81.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. fma-define 99.0%

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

   2. distribute-rgt-out 99.0%

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

   3. metadata-eval 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   4. *-commutative 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}\right)}{x} \]

6. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 99.0%

      \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]

   2. associate-/l* 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{0.375}{x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   3. sqrt-div 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \cdot \frac{0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   4. metadata-eval 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{1}}{\sqrt{x}} \cdot \frac{0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   5. frac-times 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \color{blue}{\frac{1 \cdot 0.375}{\sqrt{x} \cdot x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   6. metadata-eval 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{0.375}}{\sqrt{x} \cdot x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   7. *-commutative 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{\color{blue}{x \cdot \sqrt{x}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   8. pow1 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{\color{blue}{{x}^{1}} \cdot \sqrt{x}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   9. pow1/2 99.0%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1} \cdot \color{blue}{{x}^{0.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   10. pow-prod-up 99.0%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{\color{blue}{{x}^{\left(1 + 0.5\right)}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   11. metadata-eval 99.0%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{\color{blue}{1.5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \]

   12. *-commutative 99.0%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}\right)}{x} \]

   13. sqrt-div 98.9%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, 0.5 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)}{x} \]

   14. metadata-eval 98.9%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, 0.5 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)}{x} \]

   15. un-div-inv 98.9%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, \color{blue}{\frac{0.5}{\sqrt{x}}}\right)}{x} \]

8. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, \frac{0.5}{\sqrt{x}}\right)}{x}} \]
9. Step-by-step derivation

   1. *-lft-identity 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{0.375}{{x}^{1.5}}, \frac{0.5}{\sqrt{x}}\right)}{x}} \]

   2. fma-undefine 98.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{0.375}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}}}}{x} \]

   3. neg-mul-1 98.9%

      \[\leadsto \frac{\color{blue}{\left(-\frac{0.375}{{x}^{1.5}}\right)} + \frac{0.5}{\sqrt{x}}}{x} \]

   4. +-commutative 98.9%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{\sqrt{x}} + \left(-\frac{0.375}{{x}^{1.5}}\right)}}{x} \]

   5. distribute-neg-frac 98.9%

      \[\leadsto \frac{\frac{0.5}{\sqrt{x}} + \color{blue}{\frac{-0.375}{{x}^{1.5}}}}{x} \]

   6. metadata-eval 98.9%

      \[\leadsto \frac{\frac{0.5}{\sqrt{x}} + \frac{\color{blue}{-0.375}}{{x}^{1.5}}}{x} \]

10. Simplified 98.9%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{x}} + \frac{-0.375}{{x}^{1.5}}}{x}} \]
11. Add Preprocessing

Alternative 5: 38.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.5e+122) (pow x -0.5) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = x ** (-0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.pow(x, -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[Power[x, -0.5], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 11.5%

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

      1. frac-2neg 11.5%

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

      2. metadata-eval 11.5%

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

      3. div-inv 11.5%

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

      4. frac-2neg 11.5%

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

      5. metadata-eval 11.5%

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

      6. div-inv 11.5%

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

      7. distribute-neg-frac2 11.5%

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

      8. prod-diff 11.5%

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

      9. distribute-neg-frac 11.5%

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

      10. metadata-eval 11.5%

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

      11. +-commutative 11.5%

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

   4. Applied egg-rr 11.8%

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

   6. Simplified 11.8%

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

   7. Taylor expanded in x around 0 7.7%

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

      1. unpow1/2 7.7%

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

      2. rem-exp-log 7.7%

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

      3. exp-neg 7.7%

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

      4. exp-prod 7.7%

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

      5. distribute-lft-neg-out 7.7%

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

      6. distribute-rgt-neg-in 7.7%

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

      7. metadata-eval 7.7%

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

      8. exp-to-pow 7.7%

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

   9. Simplified 7.7%

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

   if 8.50000000000000003e122 < x

   1. Initial program 57.2%

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

      1. sub-neg 57.2%

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

      2. +-commutative 57.2%

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

      3. add-cube-cbrt 7.6%

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

      4. distribute-lft-neg-in 7.6%

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

      5. fma-define 4.6%

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

   4. Applied egg-rr 4.5%

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

   5. Taylor expanded in x around inf 57.2%

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

      1. distribute-rgt1-in 57.2%

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

      2. metadata-eval 57.2%

         \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

      3. mul0-lft 57.2%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 57.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.5 (sqrt (/ 1.0 x))) x))

C

double code(double x) {
	return (0.5 * sqrt((1.0 / x))) / x;
}

Fortran

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

Java

public static double code(double x) {
	return (0.5 * Math.sqrt((1.0 / x))) / x;
}

Python

def code(x):
	return (0.5 * math.sqrt((1.0 / x))) / x

Julia

function code(x)
	return Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * sqrt((1.0 / x))) / x;
end

Wolfram

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

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf 81.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. fma-define 99.0%

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

   2. distribute-rgt-out 99.0%

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

   3. metadata-eval 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   4. *-commutative 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}\right)}{x} \]

6. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]

7. Taylor expanded in x around inf 97.6%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]
8. Add Preprocessing

Alternative 7: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf 81.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. fma-define 99.0%

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

   2. distribute-rgt-out 99.0%

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

   3. metadata-eval 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   4. *-commutative 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}\right)}{x} \]

6. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]

7. Taylor expanded in x around inf 97.6%

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

   1. unpow1/2 97.6%

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

   2. rem-exp-log 93.6%

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

   3. exp-neg 93.6%

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

   4. exp-prod 93.6%

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

   5. distribute-lft-neg-out 93.6%

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

   6. distribute-rgt-neg-in 93.6%

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

   7. metadata-eval 93.6%

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

   8. exp-to-pow 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

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

Alternative 8: 36.7% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 37.9%

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

   1. sub-neg 37.9%

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

   2. +-commutative 37.9%

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

   3. add-cube-cbrt 9.7%

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

   4. distribute-lft-neg-in 9.7%

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

   5. fma-define 8.1%

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

4. Applied egg-rr 8.2%

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

5. Taylor expanded in x around inf 34.7%

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

   1. distribute-rgt1-in 34.7%

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

   2. metadata-eval 34.7%

      \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

   3. mul0-lft 34.7%

      \[\leadsto \color{blue}{0} \]

7. Simplified 34.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.4% 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]

Developer Target 2: 39.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} - {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return pow(x, -0.5) - pow((x + 1.0), -0.5);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024170 
(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)))))