2isqrt (example 3.6)

Percentage Accurate: 39.4% → 98.8%

Time: 13.7s

Alternatives: 7

Speedup: 1.0×

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: 39.4% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot t\_0\right) - \frac{t\_0 \cdot 0.375}{x}\right) + 0.5 \cdot \log \left(e^{{x}^{-2.5}}\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (/
    (+
     (- (fma (sqrt (/ 1.0 (pow x 5.0))) -0.25 (* 0.5 t_0)) (/ (* t_0 0.375) x))
     (* 0.5 (log (exp (pow x -2.5)))))
    x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return ((fma(sqrt((1.0 / pow(x, 5.0))), -0.25, (0.5 * t_0)) - ((t_0 * 0.375) / x)) + (0.5 * log(exp(pow(x, -2.5))))) / x;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(Float64(fma(sqrt(Float64(1.0 / (x ^ 5.0))), -0.25, Float64(0.5 * t_0)) - Float64(Float64(t_0 * 0.375) / x)) + Float64(0.5 * log(exp((x ^ -2.5))))) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.25 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 * 0.375), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Log[N[Exp[N[Power[x, -2.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in x around inf 85.3%

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

   1. *-un-lft-identity 85.3%

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

   2. pow-flip 85.3%

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

   3. sqrt-pow1 85.3%

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

   4. metadata-eval 85.3%

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

   5. metadata-eval 85.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 85.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

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

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x}} \]
11. Step-by-step derivation

    1. *-rgt-identity 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \color{blue}{\left(\sqrt{\frac{1}{{x}^{5}}} \cdot 1\right)} \cdot 0.5}{x} \]

    2. add-log-exp 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \color{blue}{\log \left(e^{\sqrt{\frac{1}{{x}^{5}}} \cdot 1}\right)} \cdot 0.5}{x} \]

    3. *-rgt-identity 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \log \left(e^{\color{blue}{\sqrt{\frac{1}{{x}^{5}}}}}\right) \cdot 0.5}{x} \]

    4. pow-flip 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \log \left(e^{\sqrt{\color{blue}{{x}^{\left(-5\right)}}}}\right) \cdot 0.5}{x} \]

    5. sqrt-pow1 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \log \left(e^{\color{blue}{{x}^{\left(\frac{-5}{2}\right)}}}\right) \cdot 0.5}{x} \]

    6. metadata-eval 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \log \left(e^{{x}^{\left(\frac{\color{blue}{-5}}{2}\right)}}\right) \cdot 0.5}{x} \]

    7. metadata-eval 98.8%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \log \left(e^{{x}^{\color{blue}{-2.5}}}\right) \cdot 0.5}{x} \]

12. Applied egg-rr 98.8%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \color{blue}{\log \left(e^{{x}^{-2.5}}\right)} \cdot 0.5}{x} \]

13. Final simplification 98.8%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + 0.5 \cdot \log \left(e^{{x}^{-2.5}}\right)}{x} \]
14. Add Preprocessing

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{\left(\mathsf{fma}\left({x}^{-2.5}, -0.25, 0.5 \cdot t\_0\right) - \frac{t\_0 \cdot 0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (/
    (+
     (- (fma (pow x -2.5) -0.25 (* 0.5 t_0)) (/ (* t_0 0.375) x))
     (* (sqrt (/ 1.0 (pow x 5.0))) 0.5))
    x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return ((fma(pow(x, -2.5), -0.25, (0.5 * t_0)) - ((t_0 * 0.375) / x)) + (sqrt((1.0 / pow(x, 5.0))) * 0.5)) / x;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(Float64(fma((x ^ -2.5), -0.25, Float64(0.5 * t_0)) - Float64(Float64(t_0 * 0.375) / x)) + Float64(sqrt(Float64(1.0 / (x ^ 5.0))) * 0.5)) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Power[x, -2.5], $MachinePrecision] * -0.25 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 * 0.375), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in x around inf 85.3%

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

   1. *-un-lft-identity 85.3%

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

   2. pow-flip 85.3%

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

   3. sqrt-pow1 85.3%

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

   4. metadata-eval 85.3%

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

   5. metadata-eval 85.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 85.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

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

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 85.3%

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

    2. pow-flip 85.3%

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

    3. sqrt-pow1 85.3%

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

    4. metadata-eval 85.3%

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

    5. metadata-eval 85.3%

       \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

12. Applied egg-rr 98.8%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{1 \cdot {x}^{-2.5}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x} \]
13. Step-by-step derivation

    1. *-lft-identity 85.3%

       \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

14. Simplified 98.8%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{{x}^{-2.5}}, -0.25, 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}\right) + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.5}{x} \]
15. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{0.5 \cdot t\_0 - \frac{t\_0 \cdot 0.375}{x}}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x)))) (/ (- (* 0.5 t_0) (/ (* t_0 0.375) x)) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return ((0.5 * t_0) - ((t_0 * 0.375) / 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 * 0.375d0) / 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 * 0.375) / x)) / x;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	return ((0.5 * t_0) - ((t_0 * 0.375) / x)) / x

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(Float64(0.5 * t_0) - Float64(Float64(t_0 * 0.375) / x)) / x)
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = ((0.5 * t_0) - ((t_0 * 0.375) / x)) / x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.5 * t$95$0), $MachinePrecision] - N[(N[(t$95$0 * 0.375), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in x around inf 85.3%

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

   1. *-un-lft-identity 85.3%

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

   2. pow-flip 85.3%

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

   3. sqrt-pow1 85.3%

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

   4. metadata-eval 85.3%

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

   5. metadata-eval 85.3%

      \[\leadsto \frac{-0.5 \cdot \left(\left(1 \cdot {x}^{\color{blue}{-2.5}}\right) \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(1 \cdot {x}^{-2.5}\right)} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-lft-identity 85.3%

      \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.3%

   \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{{x}^{-2.5}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.7%

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

   1. +-commutative 98.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 98.7%

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

   3. unsub-neg 98.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. distribute-rgt-out 98.7%

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

   5. metadata-eval 98.7%

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

10. Simplified 98.7%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}}{x}} \]
11. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \left({x}^{-2.5} - {x}^{-1.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -0.5 (- (pow x -2.5) (pow x -1.5))))

C

double code(double x) {
	return -0.5 * (pow(x, -2.5) - pow(x, -1.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.5d0) * ((x ** (-2.5d0)) - (x ** (-1.5d0)))
end function

Java

public static double code(double x) {
	return -0.5 * (Math.pow(x, -2.5) - Math.pow(x, -1.5));
}

Python

def code(x):
	return -0.5 * (math.pow(x, -2.5) - math.pow(x, -1.5))

Julia

function code(x)
	return Float64(-0.5 * Float64((x ^ -2.5) - (x ^ -1.5)))
end

MATLAB

function tmp = code(x)
	tmp = -0.5 * ((x ^ -2.5) - (x ^ -1.5));
end

Wolfram

code[x_] := N[(-0.5 * N[(N[Power[x, -2.5], $MachinePrecision] - N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.7%

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

   1. add-sqr-sqrt 20.3%

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

   2. sqrt-unprod 37.7%

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

   3. frac-times 33.4%

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

   4. metadata-eval 33.4%

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

   5. add-sqr-sqrt 29.4%

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

   6. +-commutative 29.4%

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

4. Applied egg-rr 29.4%

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

5. Taylor expanded in x around inf 84.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
6. Step-by-step derivation

   1. distribute-lft-out-- 84.4%

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

   2. associate-/l* 84.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{\frac{1}{x}} - \sqrt{x}}{{x}^{2}}} \]

   3. unpow1/2 84.4%

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

   4. rem-exp-log 84.4%

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

   5. exp-neg 84.4%

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

   6. exp-prod 84.4%

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

   7. distribute-lft-neg-out 84.4%

      \[\leadsto -0.5 \cdot \frac{e^{\color{blue}{-\log x \cdot 0.5}} - \sqrt{x}}{{x}^{2}} \]

   8. distribute-rgt-neg-in 84.4%

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

   9. metadata-eval 84.4%

      \[\leadsto -0.5 \cdot \frac{e^{\log x \cdot \color{blue}{-0.5}} - \sqrt{x}}{{x}^{2}} \]

   10. exp-to-pow 84.4%

       \[\leadsto -0.5 \cdot \frac{\color{blue}{{x}^{-0.5}} - \sqrt{x}}{{x}^{2}} \]

7. Simplified 84.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{{x}^{-0.5} - \sqrt{x}}{{x}^{2}}} \]
8. Step-by-step derivation

   1. div-sub 84.4%

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

   2. pow-div 84.4%

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

   3. metadata-eval 84.4%

      \[\leadsto -0.5 \cdot \left({x}^{\color{blue}{-2.5}} - \frac{\sqrt{x}}{{x}^{2}}\right) \]

   4. pow1/2 84.4%

      \[\leadsto -0.5 \cdot \left({x}^{-2.5} - \frac{\color{blue}{{x}^{0.5}}}{{x}^{2}}\right) \]

   5. pow-div 98.1%

      \[\leadsto -0.5 \cdot \left({x}^{-2.5} - \color{blue}{{x}^{\left(0.5 - 2\right)}}\right) \]

   6. metadata-eval 98.1%

      \[\leadsto -0.5 \cdot \left({x}^{-2.5} - {x}^{\color{blue}{-1.5}}\right) \]

9. Applied egg-rr 98.1%

   \[\leadsto -0.5 \cdot \color{blue}{\left({x}^{-2.5} - {x}^{-1.5}\right)} \]
10. Add Preprocessing

Alternative 5: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (sqrt (/ 1.0 (pow x 3.0)))))

C

double code(double x) {
	return 0.5 * sqrt((1.0 / pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
}

Python

def code(x):
	return 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
end

Wolfram

code[x_] := N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in x around inf 67.3%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
4. Add Preprocessing

Alternative 6: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{x}^{-1.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cbrt (pow x -1.5)))

C

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

Java

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

Julia

function code(x)
	return cbrt((x ^ -1.5))
end

Wolfram

code[x_] := N[Power[N[Power[x, -1.5], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in x around 0 5.8%

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

   1. add-cbrt-cube 34.9%

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

   2. add-sqr-sqrt 34.9%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}} \]

   3. sqrt-div 34.9%

      \[\leadsto \sqrt[3]{\frac{1}{x} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}} \]

   4. metadata-eval 34.9%

      \[\leadsto \sqrt[3]{\frac{1}{x} \cdot \frac{\color{blue}{1}}{\sqrt{x}}} \]

   5. frac-times 35.6%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1 \cdot 1}{x \cdot \sqrt{x}}}} \]

   6. metadata-eval 35.6%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{1}}{x \cdot \sqrt{x}}} \]

   7. add-sqr-sqrt 35.6%

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

   8. sqrt-unprod 36.3%

      \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\sqrt{x \cdot x}} \cdot \sqrt{x}}} \]

   9. sqrt-prod 36.3%

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

   10. unpow3 36.3%

       \[\leadsto \sqrt[3]{\frac{1}{\sqrt{\color{blue}{{x}^{3}}}}} \]

   11. metadata-eval 36.3%

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\sqrt{1}}}{\sqrt{{x}^{3}}}} \]

   12. sqrt-div 36.3%

       \[\leadsto \sqrt[3]{\color{blue}{\sqrt{\frac{1}{{x}^{3}}}}} \]

   13. pow1/3 36.3%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{{x}^{3}}}\right)}^{0.3333333333333333}} \]

   14. pow-flip 36.3%

       \[\leadsto {\left(\sqrt{\color{blue}{{x}^{\left(-3\right)}}}\right)}^{0.3333333333333333} \]

   15. sqrt-pow1 34.9%

       \[\leadsto {\color{blue}{\left({x}^{\left(\frac{-3}{2}\right)}\right)}}^{0.3333333333333333} \]

   16. metadata-eval 34.9%

       \[\leadsto {\left({x}^{\left(\frac{\color{blue}{-3}}{2}\right)}\right)}^{0.3333333333333333} \]

   17. metadata-eval 34.9%

       \[\leadsto {\left({x}^{\color{blue}{-1.5}}\right)}^{0.3333333333333333} \]

5. Applied egg-rr 34.9%

   \[\leadsto \color{blue}{{\left({x}^{-1.5}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 34.9%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-1.5}}} \]

7. Simplified 34.9%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-1.5}}} \]
8. Add Preprocessing

Alternative 7: 36.5% 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 37.7%

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

   1. sub-neg 37.7%

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

   2. +-commutative 37.7%

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

   3. add-cube-cbrt 12.3%

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

   4. distribute-lft-neg-in 12.3%

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

   5. fma-define 7.8%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in x around inf 34.8%

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

   1. distribute-rgt1-in 34.8%

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

   2. metadata-eval 34.8%

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

   3. mul0-lft 34.8%

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

7. Simplified 34.8%

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

Developer target: 98.5% 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 2024108 
(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)))))