2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.3%

Time: 15.6s

Alternatives: 10

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: 38.6% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{{x}^{7}}}\\ \frac{\mathsf{fma}\left({x}^{-1.5}, -0.3125, \mathsf{fma}\left(t\_0, -0.1904296875, t\_0 \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (pow x 7.0)))))
   (/
    (fma
     (pow x -1.5)
     -0.3125
     (fma
      t_0
      -0.1904296875
      (+
       (* t_0 -0.0830078125)
       (+
        (* (sqrt (/ 1.0 (pow x 5.0))) 0.3125)
        (fma (sqrt (/ 1.0 (pow x 3.0))) -0.0625 (* 0.5 (pow x -0.5)))))))
    x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / pow(x, 7.0)));
	return fma(pow(x, -1.5), -0.3125, fma(t_0, -0.1904296875, ((t_0 * -0.0830078125) + ((sqrt((1.0 / pow(x, 5.0))) * 0.3125) + fma(sqrt((1.0 / pow(x, 3.0))), -0.0625, (0.5 * pow(x, -0.5))))))) / x;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / (x ^ 7.0)))
	return Float64(fma((x ^ -1.5), -0.3125, fma(t_0, -0.1904296875, Float64(Float64(t_0 * -0.0830078125) + Float64(Float64(sqrt(Float64(1.0 / (x ^ 5.0))) * 0.3125) + fma(sqrt(Float64(1.0 / (x ^ 3.0))), -0.0625, Float64(0.5 * (x ^ -0.5))))))) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[x, -1.5], $MachinePrecision] * -0.3125 + N[(t$95$0 * -0.1904296875 + N[(N[(t$95$0 * -0.0830078125), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3125), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.0625 + N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 37.6%

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

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. div-inv 37.6%

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

   4. frac-2neg 37.6%

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

   5. metadata-eval 37.6%

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

   6. div-inv 37.6%

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

   7. distribute-neg-frac2 37.6%

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

   8. prod-diff 37.6%

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

   9. distribute-neg-frac 37.6%

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

   10. metadata-eval 37.6%

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

   11. +-commutative 37.6%

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

4. Applied egg-rr 31.2%

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

6. Simplified 31.2%

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

   1. inv-pow 31.6%

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

   2. sqrt-pow2 37.7%

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

   3. metadata-eval 37.7%

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

   4. metadata-eval 37.7%

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

   5. pow-pow 22.2%

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

   6. sqr-pow 19.1%

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

   7. sqr-pow 27.1%

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

   8. difference-of-squares 27.0%

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

   9. metadata-eval 27.0%

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

   10. pow-pow 27.0%

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

   11. metadata-eval 27.0%

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

   12. metadata-eval 27.0%

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

   13. metadata-eval 27.0%

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

   14. pow-pow 37.6%

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

   15. metadata-eval 37.6%

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

   16. metadata-eval 37.6%

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

8. Applied egg-rr 37.6%

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

9. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{\frac{-0.3125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.1904296875 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(-0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.05859375 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(-0.0244140625 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(0.078125 \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(0.234375 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)\right)\right)\right)}{x}} \]

11. Simplified 99.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{1 \cdot \sqrt{\frac{1}{{x}^{3}}}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

    2. pow-flip 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(1 \cdot \sqrt{\color{blue}{{x}^{\left(-3\right)}}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

    3. sqrt-pow1 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(1 \cdot \color{blue}{{x}^{\left(\frac{-3}{2}\right)}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

    4. metadata-eval 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(1 \cdot {x}^{\left(\frac{\color{blue}{-3}}{2}\right)}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

    5. metadata-eval 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(1 \cdot {x}^{\color{blue}{-1.5}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

13. Applied egg-rr 99.1%

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{1 \cdot {x}^{-1.5}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]
14. Step-by-step derivation

    1. *-lft-identity 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{-1.5}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

15. Simplified 99.1%

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{-1.5}}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]

16. Final simplification 99.1%

    \[\leadsto \frac{\mathsf{fma}\left({x}^{-1.5}, -0.3125, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{7}}}, -0.1904296875, \sqrt{\frac{1}{{x}^{7}}} \cdot -0.0830078125 + \left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, -0.0625, 0.5 \cdot {x}^{-0.5}\right)\right)\right)\right)}{x} \]
17. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + 0.5 \cdot {x}^{-0.5}\right) + \sqrt{\frac{1}{{x}^{3}}} \cdot -0.375}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (+ (* (sqrt (/ 1.0 (pow x 5.0))) 0.3125) (* 0.5 (pow x -0.5)))
   (* (sqrt (/ 1.0 (pow x 3.0))) -0.375))
  x))

C

double code(double x) {
	return (((sqrt((1.0 / pow(x, 5.0))) * 0.3125) + (0.5 * pow(x, -0.5))) + (sqrt((1.0 / pow(x, 3.0))) * -0.375)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((sqrt((1.0d0 / (x ** 5.0d0))) * 0.3125d0) + (0.5d0 * (x ** (-0.5d0)))) + (sqrt((1.0d0 / (x ** 3.0d0))) * (-0.375d0))) / x
end function

Java

public static double code(double x) {
	return (((Math.sqrt((1.0 / Math.pow(x, 5.0))) * 0.3125) + (0.5 * Math.pow(x, -0.5))) + (Math.sqrt((1.0 / Math.pow(x, 3.0))) * -0.375)) / x;
}

Python

def code(x):
	return (((math.sqrt((1.0 / math.pow(x, 5.0))) * 0.3125) + (0.5 * math.pow(x, -0.5))) + (math.sqrt((1.0 / math.pow(x, 3.0))) * -0.375)) / x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / (x ^ 5.0))) * 0.3125) + Float64(0.5 * (x ^ -0.5))) + Float64(sqrt(Float64(1.0 / (x ^ 3.0))) * -0.375)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((sqrt((1.0 / (x ^ 5.0))) * 0.3125) + (0.5 * (x ^ -0.5))) + (sqrt((1.0 / (x ^ 3.0))) * -0.375)) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3125), $MachinePrecision] + N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. div-inv 37.6%

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

   4. frac-2neg 37.6%

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

   5. metadata-eval 37.6%

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

   6. div-inv 37.6%

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

   7. distribute-neg-frac2 37.6%

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

   8. prod-diff 37.6%

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

   9. distribute-neg-frac 37.6%

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

   10. metadata-eval 37.6%

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

   11. +-commutative 37.6%

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

4. Applied egg-rr 31.2%

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

6. Simplified 31.2%

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

   1. inv-pow 31.6%

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

   2. sqrt-pow2 37.7%

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

   3. metadata-eval 37.7%

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

   4. metadata-eval 37.7%

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

   5. pow-pow 22.2%

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

   6. sqr-pow 19.1%

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

   7. sqr-pow 27.1%

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

   8. difference-of-squares 27.0%

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

   9. metadata-eval 27.0%

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

   10. pow-pow 27.0%

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

   11. metadata-eval 27.0%

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

   12. metadata-eval 27.0%

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

   13. metadata-eval 27.0%

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

   14. pow-pow 37.6%

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

   15. metadata-eval 37.6%

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

   16. metadata-eval 37.6%

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

8. Applied egg-rr 37.6%

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

9. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{-0.3125 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.078125 \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(0.234375 \cdot \sqrt{\frac{1}{{x}^{5}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{x}} \]
10. Step-by-step derivation

11. Simplified 98.9%

    \[\leadsto \color{blue}{\frac{\left(0.5 \cdot {x}^{-0.5} + \sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125\right) + \sqrt{\frac{1}{{x}^{3}}} \cdot -0.375}{x}} \]

12. Final simplification 98.9%

    \[\leadsto \frac{\left(\sqrt{\frac{1}{{x}^{5}}} \cdot 0.3125 + 0.5 \cdot {x}^{-0.5}\right) + \sqrt{\frac{1}{{x}^{3}}} \cdot -0.375}{x} \]
13. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around inf 80.2%

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

   1. *-un-lft-identity 80.2%

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

   2. pow1/2 80.2%

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

   3. pow-flip 80.2%

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

   4. pow-pow 80.2%

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

   5. metadata-eval 80.2%

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

   6. metadata-eval 80.2%

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

5. Applied egg-rr 80.2%

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

   1. *-lft-identity 80.2%

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

7. Simplified 80.2%

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

8. Taylor expanded in x around inf 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around inf 78.5%

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

   1. distribute-lft-out-- 78.5%

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

5. Simplified 78.5%

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

   1. *-commutative 78.5%

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

   2. unpow2 78.5%

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

   3. times-frac 96.8%

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

   4. inv-pow 96.8%

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

   5. sqrt-pow1 96.8%

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

   6. metadata-eval 96.8%

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

7. Applied egg-rr 96.8%

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

8. Final simplification 96.8%

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

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot \left(-\sqrt{x}\right)}{{x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* -0.5 (- (sqrt x))) (pow x 2.0)))

C

double code(double x) {
	return (-0.5 * -sqrt(x)) / pow(x, 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-0.5d0) * -sqrt(x)) / (x ** 2.0d0)
end function

Java

public static double code(double x) {
	return (-0.5 * -Math.sqrt(x)) / Math.pow(x, 2.0);
}

Python

def code(x):
	return (-0.5 * -math.sqrt(x)) / math.pow(x, 2.0)

Julia

function code(x)
	return Float64(Float64(-0.5 * Float64(-sqrt(x))) / (x ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (-0.5 * -sqrt(x)) / (x ^ 2.0);
end

Wolfram

code[x_] := N[(N[(-0.5 * (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around inf 78.5%

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

   1. distribute-lft-out-- 78.5%

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

5. Simplified 78.5%

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

6. Taylor expanded in x around inf 78.4%

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

   1. neg-mul-1 78.4%

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

8. Simplified 78.4%

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

9. Final simplification 78.4%

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

Alternative 6: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0.125 \cdot {x}^{-4.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cbrt (* 0.125 (pow x -4.5))))

C

double code(double x) {
	return cbrt((0.125 * pow(x, -4.5)));
}

Java

public static double code(double x) {
	return Math.cbrt((0.125 * Math.pow(x, -4.5)));
}

Julia

function code(x)
	return cbrt(Float64(0.125 * (x ^ -4.5)))
end

Wolfram

code[x_] := N[Power[N[(0.125 * N[Power[x, -4.5], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 37.6%

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

   1. add-cbrt-cube 37.6%

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

   2. pow3 37.6%

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

   3. inv-pow 37.6%

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

   4. sqrt-pow2 37.6%

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

   5. metadata-eval 37.6%

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

   6. pow1/2 37.6%

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

   7. pow-flip 37.7%

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

   8. +-commutative 37.7%

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

   9. metadata-eval 37.7%

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

4. Applied egg-rr 37.7%

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

5. Taylor expanded in x around inf 42.2%

   \[\leadsto \sqrt[3]{\color{blue}{0.125 \cdot \sqrt{\frac{1}{{x}^{9}}}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 42.2%

      \[\leadsto \color{blue}{1 \cdot \sqrt[3]{0.125 \cdot \sqrt{\frac{1}{{x}^{9}}}}} \]

   2. pow1/2 42.2%

      \[\leadsto 1 \cdot \sqrt[3]{0.125 \cdot \color{blue}{{\left(\frac{1}{{x}^{9}}\right)}^{0.5}}} \]

   3. pow-flip 42.2%

      \[\leadsto 1 \cdot \sqrt[3]{0.125 \cdot {\color{blue}{\left({x}^{\left(-9\right)}\right)}}^{0.5}} \]

   4. pow-pow 52.8%

      \[\leadsto 1 \cdot \sqrt[3]{0.125 \cdot \color{blue}{{x}^{\left(\left(-9\right) \cdot 0.5\right)}}} \]

   5. metadata-eval 52.8%

      \[\leadsto 1 \cdot \sqrt[3]{0.125 \cdot {x}^{\left(\color{blue}{-9} \cdot 0.5\right)}} \]

   6. metadata-eval 52.8%

      \[\leadsto 1 \cdot \sqrt[3]{0.125 \cdot {x}^{\color{blue}{-4.5}}} \]

7. Applied egg-rr 52.8%

   \[\leadsto \color{blue}{1 \cdot \sqrt[3]{0.125 \cdot {x}^{-4.5}}} \]
8. Step-by-step derivation

   1. *-lft-identity 52.8%

      \[\leadsto \color{blue}{\sqrt[3]{0.125 \cdot {x}^{-4.5}}} \]

9. Simplified 52.8%

   \[\leadsto \color{blue}{\sqrt[3]{0.125 \cdot {x}^{-4.5}}} \]

10. Final simplification 52.8%

    \[\leadsto \sqrt[3]{0.125 \cdot {x}^{-4.5}} \]
11. Add Preprocessing

Alternative 7: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around inf 78.5%

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

   1. distribute-lft-out-- 78.5%

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

5. Simplified 78.5%

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

6. Taylor expanded in x around inf 62.0%

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

   1. exp-to-pow 59.8%

      \[\leadsto 0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \]

   2. *-commutative 59.8%

      \[\leadsto 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]

   3. rec-exp 61.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \]

   4. distribute-lft-neg-in 61.9%

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

   5. metadata-eval 61.9%

      \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

   6. *-commutative 61.9%

      \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

   7. exp-to-pow 64.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

8. Simplified 64.1%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{{x}^{-3}}} \]

9. Final simplification 64.1%

   \[\leadsto 0.5 \cdot \sqrt{{x}^{-3}} \]
10. Add Preprocessing

Alternative 8: 37.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left({x}^{2}\right)}^{-0.25} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (pow x 2.0) -0.25))

C

double code(double x) {
	return pow(pow(x, 2.0), -0.25);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) ** (-0.25d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow(x, 2.0), -0.25);
}

Python

def code(x):
	return math.pow(math.pow(x, 2.0), -0.25)

Julia

function code(x)
	return (x ^ 2.0) ^ -0.25
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) ^ -0.25;
end

Wolfram

code[x_] := N[Power[N[Power[x, 2.0], $MachinePrecision], -0.25], $MachinePrecision]

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around 0 5.7%

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

   1. inv-pow 5.7%

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

   2. sqrt-pow1 5.7%

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

   3. metadata-eval 5.7%

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

   4. sqr-pow 5.7%

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

   5. pow-prod-down 34.8%

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

   6. pow2 34.8%

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

   7. metadata-eval 34.8%

      \[\leadsto {\left({x}^{2}\right)}^{\color{blue}{-0.25}} \]

5. Applied egg-rr 34.8%

   \[\leadsto \color{blue}{{\left({x}^{2}\right)}^{-0.25}} \]

6. Final simplification 34.8%

   \[\leadsto {\left({x}^{2}\right)}^{-0.25} \]
7. Add Preprocessing

Alternative 9: 34.4% 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 37.6%

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

3. Taylor expanded in x around 0 5.7%

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

   1. add-cbrt-cube 32.2%

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

   2. pow1/3 32.2%

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

   3. add-sqr-sqrt 32.2%

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

   4. pow1 32.2%

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

   5. pow1/2 32.2%

      \[\leadsto {\left({\left(\frac{1}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

   6. pow-prod-up 32.2%

      \[\leadsto {\color{blue}{\left({\left(\frac{1}{x}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

   7. metadata-eval 32.2%

      \[\leadsto {\left({\left(\frac{1}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   8. metadata-eval 32.2%

      \[\leadsto {\left({\left(\frac{1}{x}\right)}^{\color{blue}{\left(0.5 \cdot 3\right)}}\right)}^{0.3333333333333333} \]

   9. pow-pow 32.2%

      \[\leadsto {\color{blue}{\left({\left({\left(\frac{1}{x}\right)}^{0.5}\right)}^{3}\right)}}^{0.3333333333333333} \]

   10. inv-pow 32.2%

       \[\leadsto {\left({\left({\color{blue}{\left({x}^{-1}\right)}}^{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

   11. pow-pow 32.2%

       \[\leadsto {\left({\color{blue}{\left({x}^{\left(-1 \cdot 0.5\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   12. metadata-eval 32.2%

       \[\leadsto {\left({\left({x}^{\color{blue}{-0.5}}\right)}^{3}\right)}^{0.3333333333333333} \]

   13. pow-pow 32.2%

       \[\leadsto {\color{blue}{\left({x}^{\left(-0.5 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   14. metadata-eval 32.2%

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

5. Applied egg-rr 32.2%

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

   1. unpow1/3 32.2%

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

7. Simplified 32.2%

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

8. Final simplification 32.2%

   \[\leadsto \sqrt[3]{{x}^{-1.5}} \]
9. Add Preprocessing

Alternative 10: 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 37.6%

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

3. Taylor expanded in x around 0 5.7%

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

   1. inv-pow 5.7%

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

   2. sqrt-pow1 5.7%

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

   3. metadata-eval 5.7%

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

   4. *-un-lft-identity 5.7%

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

5. Applied egg-rr 5.7%

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

   1. *-lft-identity 5.7%

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

7. Simplified 5.7%

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

8. Final simplification 5.7%

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

Developer target: 98.3% 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 2024076 
(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)))))