2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.6%

Time: 9.4s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.7% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

   1. flip-+ 38.8%

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

   2. sqrt-undiv 38.9%

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

   3. sqrt-undiv 38.8%

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

   4. add-sqr-sqrt 39.0%

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

   5. metadata-eval 39.0%

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

   6. sqrt-undiv 39.0%

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

8. Applied egg-rr 39.0%

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

   1. *-rgt-identity 39.0%

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

   2. associate-*r/ 32.4%

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

   3. *-commutative 32.4%

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

   4. distribute-rgt-in 32.3%

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

   5. rgt-mult-inverse 38.9%

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

   6. *-lft-identity 38.9%

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

   7. +-commutative 38.9%

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

   8. rem-exp-log 38.9%

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

   9. log1p-undefine 38.9%

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}} - 1}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   10. expm1-define 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   11. sub-neg 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} + \left(--1\right)}} \cdot {\left(1 + x\right)}^{-0.5} \]

   12. *-lft-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 \cdot \sqrt{\frac{1 + x}{x}}} + \left(--1\right)} \cdot {\left(1 + x\right)}^{-0.5} \]

   13. metadata-eval 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 \cdot \sqrt{\frac{1 + x}{x}} + \color{blue}{1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   14. *-commutative 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} \cdot 1} + 1} \cdot {\left(1 + x\right)}^{-0.5} \]

   15. distribute-lft1-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\left(\sqrt{\frac{1 + x}{x}} + 1\right) \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   16. distribute-rgt1-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 + \sqrt{\frac{1 + x}{x}} \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   17. *-rgt-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \color{blue}{\sqrt{\frac{1 + x}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   18. *-rgt-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\frac{\color{blue}{\left(1 + x\right) \cdot 1}}{x}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   19. associate-*r/ 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\left(1 + x\right) \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   20. *-commutative 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\frac{1}{x} \cdot \left(1 + x\right)}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   21. distribute-rgt-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{1 \cdot \frac{1}{x} + x \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{1 + \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

11. Taylor expanded in x around inf 99.7%

    \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{1 + \frac{1}{x}}} \cdot {\color{blue}{\left(x \cdot \left(1 + \frac{1}{x}\right)\right)}}^{-0.5} \]

12. Taylor expanded in x around 0 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

   1. flip-+ 38.8%

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

   2. sqrt-undiv 38.9%

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

   3. sqrt-undiv 38.8%

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

   4. add-sqr-sqrt 39.0%

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

   5. metadata-eval 39.0%

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

   6. sqrt-undiv 39.0%

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

8. Applied egg-rr 39.0%

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

   1. *-rgt-identity 39.0%

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

   2. associate-*r/ 32.4%

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

   3. *-commutative 32.4%

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

   4. distribute-rgt-in 32.3%

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

   5. rgt-mult-inverse 38.9%

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

   6. *-lft-identity 38.9%

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

   7. +-commutative 38.9%

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

   8. rem-exp-log 38.9%

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

   9. log1p-undefine 38.9%

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}} - 1}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   10. expm1-define 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   11. sub-neg 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} + \left(--1\right)}} \cdot {\left(1 + x\right)}^{-0.5} \]

   12. *-lft-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 \cdot \sqrt{\frac{1 + x}{x}}} + \left(--1\right)} \cdot {\left(1 + x\right)}^{-0.5} \]

   13. metadata-eval 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 \cdot \sqrt{\frac{1 + x}{x}} + \color{blue}{1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   14. *-commutative 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} \cdot 1} + 1} \cdot {\left(1 + x\right)}^{-0.5} \]

   15. distribute-lft1-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\left(\sqrt{\frac{1 + x}{x}} + 1\right) \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   16. distribute-rgt1-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 + \sqrt{\frac{1 + x}{x}} \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   17. *-rgt-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \color{blue}{\sqrt{\frac{1 + x}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   18. *-rgt-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\frac{\color{blue}{\left(1 + x\right) \cdot 1}}{x}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   19. associate-*r/ 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\left(1 + x\right) \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   20. *-commutative 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\frac{1}{x} \cdot \left(1 + x\right)}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   21. distribute-rgt-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{1 \cdot \frac{1}{x} + x \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{1 + \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

11. Taylor expanded in x around 0 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ {\left(1 + x\right)}^{-0.5} \cdot \frac{\frac{\frac{0.0625 + 0.0390625 \cdot \frac{-1}{x}}{x} - 0.125}{x} + 0.5}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (+ 1.0 x) -0.5)
  (/ (+ (/ (- (/ (+ 0.0625 (* 0.0390625 (/ -1.0 x))) x) 0.125) x) 0.5) x)))

C

double code(double x) {
	return pow((1.0 + x), -0.5) * ((((((0.0625 + (0.0390625 * (-1.0 / x))) / x) - 0.125) / x) + 0.5) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + x) ** (-0.5d0)) * ((((((0.0625d0 + (0.0390625d0 * ((-1.0d0) / x))) / x) - 0.125d0) / x) + 0.5d0) / x)
end function

Java

public static double code(double x) {
	return Math.pow((1.0 + x), -0.5) * ((((((0.0625 + (0.0390625 * (-1.0 / x))) / x) - 0.125) / x) + 0.5) / x);
}

Python

def code(x):
	return math.pow((1.0 + x), -0.5) * ((((((0.0625 + (0.0390625 * (-1.0 / x))) / x) - 0.125) / x) + 0.5) / x)

Julia

function code(x)
	return Float64((Float64(1.0 + x) ^ -0.5) * Float64(Float64(Float64(Float64(Float64(Float64(0.0625 + Float64(0.0390625 * Float64(-1.0 / x))) / x) - 0.125) / x) + 0.5) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + x) ^ -0.5) * ((((((0.0625 + (0.0390625 * (-1.0 / x))) / x) - 0.125) / x) + 0.5) / x);
end

Wolfram

code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(N[(N[(N[(N[(0.0625 + N[(0.0390625 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.125), $MachinePrecision] / x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

   1. flip-+ 38.8%

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

   2. sqrt-undiv 38.9%

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

   3. sqrt-undiv 38.8%

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

   4. add-sqr-sqrt 39.0%

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

   5. metadata-eval 39.0%

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

   6. sqrt-undiv 39.0%

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

8. Applied egg-rr 39.0%

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

   1. *-rgt-identity 39.0%

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

   2. associate-*r/ 32.4%

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

   3. *-commutative 32.4%

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

   4. distribute-rgt-in 32.3%

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

   5. rgt-mult-inverse 38.9%

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

   6. *-lft-identity 38.9%

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

   7. +-commutative 38.9%

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

   8. rem-exp-log 38.9%

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

   9. log1p-undefine 38.9%

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}} - 1}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   10. expm1-define 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}}{\sqrt{\frac{1 + x}{x}} - -1} \cdot {\left(1 + x\right)}^{-0.5} \]

   11. sub-neg 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} + \left(--1\right)}} \cdot {\left(1 + x\right)}^{-0.5} \]

   12. *-lft-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 \cdot \sqrt{\frac{1 + x}{x}}} + \left(--1\right)} \cdot {\left(1 + x\right)}^{-0.5} \]

   13. metadata-eval 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 \cdot \sqrt{\frac{1 + x}{x}} + \color{blue}{1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   14. *-commutative 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\sqrt{\frac{1 + x}{x}} \cdot 1} + 1} \cdot {\left(1 + x\right)}^{-0.5} \]

   15. distribute-lft1-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{\left(\sqrt{\frac{1 + x}{x}} + 1\right) \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   16. distribute-rgt1-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{\color{blue}{1 + \sqrt{\frac{1 + x}{x}} \cdot 1}} \cdot {\left(1 + x\right)}^{-0.5} \]

   17. *-rgt-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \color{blue}{\sqrt{\frac{1 + x}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   18. *-rgt-identity 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\frac{\color{blue}{\left(1 + x\right) \cdot 1}}{x}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   19. associate-*r/ 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\left(1 + x\right) \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   20. *-commutative 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{\frac{1}{x} \cdot \left(1 + x\right)}}} \cdot {\left(1 + x\right)}^{-0.5} \]

   21. distribute-rgt-in 99.7%

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{\color{blue}{1 \cdot \frac{1}{x} + x \cdot \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)}{1 + \sqrt{1 + \frac{1}{x}}}} \cdot {\left(1 + x\right)}^{-0.5} \]

11. Taylor expanded in x around -inf 99.0%

    \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.0390625 \cdot \frac{1}{x} - 0.0625}{x} - 0.125}{x} - 0.5}{x}\right)} \cdot {\left(1 + x\right)}^{-0.5} \]

12. Final simplification 99.0%

    \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{\frac{\frac{0.0625 + 0.0390625 \cdot \frac{-1}{x}}{x} - 0.125}{x} + 0.5}{x} \]
13. Add Preprocessing

Alternative 4: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(1 + x\right)}^{-0.5}}{\frac{x}{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (pow (+ 1.0 x) -0.5) (/ x (+ 0.5 (/ (+ (/ 0.0625 x) -0.125) x)))))

C

double code(double x) {
	return pow((1.0 + x), -0.5) / (x / (0.5 + (((0.0625 / x) + -0.125) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + x) ** (-0.5d0)) / (x / (0.5d0 + (((0.0625d0 / x) + (-0.125d0)) / x)))
end function

Java

public static double code(double x) {
	return Math.pow((1.0 + x), -0.5) / (x / (0.5 + (((0.0625 / x) + -0.125) / x)));
}

Python

def code(x):
	return math.pow((1.0 + x), -0.5) / (x / (0.5 + (((0.0625 / x) + -0.125) / x)))

Julia

function code(x)
	return Float64((Float64(1.0 + x) ^ -0.5) / Float64(x / Float64(0.5 + Float64(Float64(Float64(0.0625 / x) + -0.125) / x))))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + x) ^ -0.5) / (x / (0.5 + (((0.0625 / x) + -0.125) / x)));
end

Wolfram

code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] / N[(x / N[(0.5 + N[(N[(N[(0.0625 / x), $MachinePrecision] + -0.125), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate--l+ 98.9%

      \[\leadsto \frac{\color{blue}{0.5 + \left(\frac{0.0625}{{x}^{2}} - 0.125 \cdot \frac{1}{x}\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. associate-*r/ 98.9%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{\color{blue}{0.125}}{x}\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
10. Step-by-step derivation

    1. *-commutative 98.9%

       \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}} \]

    2. add-exp-log 93.4%

       \[\leadsto \color{blue}{e^{\log \left({\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)}} \]

    3. log-prod 92.9%

       \[\leadsto e^{\color{blue}{\log \left({\left(1 + x\right)}^{-0.5}\right) + \log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)}} \]

    4. log-pow 92.9%

       \[\leadsto e^{\color{blue}{-0.5 \cdot \log \left(1 + x\right)} + \log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)} \]

    5. log1p-define 92.9%

       \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(x\right)} + \log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)} \]

    6. associate-+r- 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\color{blue}{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \frac{0.125}{x}}}{x}\right)} \]

    7. add-sqr-sqrt 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + \color{blue}{\sqrt{\frac{0.0625}{{x}^{2}}} \cdot \sqrt{\frac{0.0625}{{x}^{2}}}}\right) - \frac{0.125}{x}}{x}\right)} \]

    8. pow2 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + \color{blue}{{\left(\sqrt{\frac{0.0625}{{x}^{2}}}\right)}^{2}}\right) - \frac{0.125}{x}}{x}\right)} \]

    9. sqrt-div 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\color{blue}{\left(\frac{\sqrt{0.0625}}{\sqrt{{x}^{2}}}\right)}}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    10. metadata-eval 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{\color{blue}{0.25}}{\sqrt{{x}^{2}}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    11. sqrt-pow1 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    12. metadata-eval 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{{x}^{\color{blue}{1}}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    13. pow1 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{\color{blue}{x}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

11. Applied egg-rr 92.9%

    \[\leadsto \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)}} \]
12. Step-by-step derivation

    1. exp-sum 93.4%

       \[\leadsto \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)}} \]

    2. *-commutative 93.4%

       \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot -0.5}} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    3. log1p-undefine 93.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + x\right)} \cdot -0.5} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    4. exp-to-pow 93.8%

       \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5}} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    5. rem-exp-log 98.9%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \color{blue}{\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}} \]

13. Simplified 98.9%

    \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}{x}} \]
14. Step-by-step derivation

    1. clear-num 98.9%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \color{blue}{\frac{1}{\frac{x}{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}}} \]

    2. un-div-inv 99.0%

       \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{-0.5}}{\frac{x}{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}}} \]

15. Applied egg-rr 99.0%

    \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{-0.5}}{\frac{x}{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}}} \]
16. Add Preprocessing

Alternative 5: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (+ 0.5 (/ (+ (/ 0.0625 x) -0.125) x)) (/ (pow (+ 1.0 x) -0.5) x)))

C

double code(double x) {
	return (0.5 + (((0.0625 / x) + -0.125) / x)) * (pow((1.0 + x), -0.5) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + (((0.0625d0 / x) + (-0.125d0)) / x)) * (((1.0d0 + x) ** (-0.5d0)) / x)
end function

Java

public static double code(double x) {
	return (0.5 + (((0.0625 / x) + -0.125) / x)) * (Math.pow((1.0 + x), -0.5) / x);
}

Python

def code(x):
	return (0.5 + (((0.0625 / x) + -0.125) / x)) * (math.pow((1.0 + x), -0.5) / x)

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(Float64(Float64(0.0625 / x) + -0.125) / x)) * Float64((Float64(1.0 + x) ^ -0.5) / x))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (((0.0625 / x) + -0.125) / x)) * (((1.0 + x) ^ -0.5) / x);
end

Wolfram

code[x_] := N[(N[(0.5 + N[(N[(N[(0.0625 / x), $MachinePrecision] + -0.125), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate--l+ 98.9%

      \[\leadsto \frac{\color{blue}{0.5 + \left(\frac{0.0625}{{x}^{2}} - 0.125 \cdot \frac{1}{x}\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. associate-*r/ 98.9%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{\color{blue}{0.125}}{x}\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
10. Step-by-step derivation

    1. associate-*l/ 99.0%

       \[\leadsto \color{blue}{\frac{\left(0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)\right) \cdot {\left(1 + x\right)}^{-0.5}}{x}} \]

    2. associate-+r- 99.0%

       \[\leadsto \frac{\color{blue}{\left(\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \frac{0.125}{x}\right)} \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    3. add-sqr-sqrt 99.0%

       \[\leadsto \frac{\left(\left(0.5 + \color{blue}{\sqrt{\frac{0.0625}{{x}^{2}}} \cdot \sqrt{\frac{0.0625}{{x}^{2}}}}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    4. pow2 99.0%

       \[\leadsto \frac{\left(\left(0.5 + \color{blue}{{\left(\sqrt{\frac{0.0625}{{x}^{2}}}\right)}^{2}}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    5. sqrt-div 99.0%

       \[\leadsto \frac{\left(\left(0.5 + {\color{blue}{\left(\frac{\sqrt{0.0625}}{\sqrt{{x}^{2}}}\right)}}^{2}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    6. metadata-eval 99.0%

       \[\leadsto \frac{\left(\left(0.5 + {\left(\frac{\color{blue}{0.25}}{\sqrt{{x}^{2}}}\right)}^{2}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    7. sqrt-pow1 99.0%

       \[\leadsto \frac{\left(\left(0.5 + {\left(\frac{0.25}{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}\right)}^{2}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    8. metadata-eval 99.0%

       \[\leadsto \frac{\left(\left(0.5 + {\left(\frac{0.25}{{x}^{\color{blue}{1}}}\right)}^{2}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

    9. pow1 99.0%

       \[\leadsto \frac{\left(\left(0.5 + {\left(\frac{0.25}{\color{blue}{x}}\right)}^{2}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x} \]

11. Applied egg-rr 99.0%

    \[\leadsto \color{blue}{\frac{\left(\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{x}} \]
12. Step-by-step derivation

    1. associate-/l* 99.0%

       \[\leadsto \color{blue}{\left(\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{x}} \]

13. Simplified 98.9%

    \[\leadsto \color{blue}{\left(0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{x}} \]
14. Add Preprocessing

Alternative 6: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 - \frac{0.125}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (+ 1.0 x) -0.5) (/ (- 0.5 (/ 0.125 x)) x)))

C

double code(double x) {
	return pow((1.0 + x), -0.5) * ((0.5 - (0.125 / x)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + x) ** (-0.5d0)) * ((0.5d0 - (0.125d0 / x)) / x)
end function

Java

public static double code(double x) {
	return Math.pow((1.0 + x), -0.5) * ((0.5 - (0.125 / x)) / x);
}

Python

def code(x):
	return math.pow((1.0 + x), -0.5) * ((0.5 - (0.125 / x)) / x)

Julia

function code(x)
	return Float64((Float64(1.0 + x) ^ -0.5) * Float64(Float64(0.5 - Float64(0.125 / x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + x) ^ -0.5) * ((0.5 - (0.125 / x)) / x);
end

Wolfram

code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

7. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

    \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 - \frac{0.125}{x}}{x} \]
11. Add Preprocessing

Alternative 7: 98.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. frac-2neg 38.8%

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

   2. metadata-eval 38.8%

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

   3. div-inv 38.8%

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

   4. frac-2neg 38.8%

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

   5. metadata-eval 38.8%

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

   6. div-inv 38.8%

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

   7. distribute-neg-frac2 38.8%

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

   8. prod-diff 38.8%

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

   9. distribute-neg-frac 38.8%

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

   10. metadata-eval 38.8%

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

   11. +-commutative 38.8%

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

4. Applied egg-rr 30.0%

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

   1. +-commutative 30.0%

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

   2. fma-undefine 30.0%

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

   3. mul-1-neg 30.0%

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

   4. distribute-neg-out 30.0%

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

   5. unsub-neg 30.0%

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

6. Simplified 30.0%

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

7. Taylor expanded in x around inf 63.6%

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

   1. *-commutative 63.6%

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

9. Simplified 63.6%

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

    1. *-un-lft-identity 63.6%

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

    2. pow-flip 64.0%

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

    3. sqrt-pow1 98.1%

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

    4. metadata-eval 98.1%

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

    5. metadata-eval 98.1%

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

11. Applied egg-rr 98.1%

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

    1. *-lft-identity 98.1%

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

13. Simplified 98.1%

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

14. Final simplification 98.1%

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

Alternative 8: 7.9% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ 0.5 (/ (+ (/ 0.0625 x) -0.125) x)) x))

C

double code(double x) {
	return (0.5 + (((0.0625 / x) + -0.125) / x)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + (((0.0625d0 / x) + (-0.125d0)) / x)) / x
end function

Java

public static double code(double x) {
	return (0.5 + (((0.0625 / x) + -0.125) / x)) / x;
}

Python

def code(x):
	return (0.5 + (((0.0625 / x) + -0.125) / x)) / x

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(Float64(Float64(0.0625 / x) + -0.125) / x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (((0.0625 / x) + -0.125) / x)) / x;
end

Wolfram

code[x_] := N[(N[(0.5 + N[(N[(N[(0.0625 / x), $MachinePrecision] + -0.125), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate--l+ 98.9%

      \[\leadsto \frac{\color{blue}{0.5 + \left(\frac{0.0625}{{x}^{2}} - 0.125 \cdot \frac{1}{x}\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. associate-*r/ 98.9%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{\color{blue}{0.125}}{x}\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
10. Step-by-step derivation

    1. *-commutative 98.9%

       \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}} \]

    2. add-exp-log 93.4%

       \[\leadsto \color{blue}{e^{\log \left({\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)}} \]

    3. log-prod 92.9%

       \[\leadsto e^{\color{blue}{\log \left({\left(1 + x\right)}^{-0.5}\right) + \log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)}} \]

    4. log-pow 92.9%

       \[\leadsto e^{\color{blue}{-0.5 \cdot \log \left(1 + x\right)} + \log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)} \]

    5. log1p-define 92.9%

       \[\leadsto e^{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(x\right)} + \log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \frac{0.125}{x}\right)}{x}\right)} \]

    6. associate-+r- 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\color{blue}{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \frac{0.125}{x}}}{x}\right)} \]

    7. add-sqr-sqrt 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + \color{blue}{\sqrt{\frac{0.0625}{{x}^{2}}} \cdot \sqrt{\frac{0.0625}{{x}^{2}}}}\right) - \frac{0.125}{x}}{x}\right)} \]

    8. pow2 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + \color{blue}{{\left(\sqrt{\frac{0.0625}{{x}^{2}}}\right)}^{2}}\right) - \frac{0.125}{x}}{x}\right)} \]

    9. sqrt-div 92.9%

       \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\color{blue}{\left(\frac{\sqrt{0.0625}}{\sqrt{{x}^{2}}}\right)}}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    10. metadata-eval 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{\color{blue}{0.25}}{\sqrt{{x}^{2}}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    11. sqrt-pow1 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    12. metadata-eval 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{{x}^{\color{blue}{1}}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    13. pow1 92.9%

        \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{\color{blue}{x}}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

11. Applied egg-rr 92.9%

    \[\leadsto \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right) + \log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)}} \]
12. Step-by-step derivation

    1. exp-sum 93.4%

       \[\leadsto \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)}} \]

    2. *-commutative 93.4%

       \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot -0.5}} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    3. log1p-undefine 93.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + x\right)} \cdot -0.5} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    4. exp-to-pow 93.8%

       \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5}} \cdot e^{\log \left(\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}\right)} \]

    5. rem-exp-log 98.9%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \color{blue}{\frac{\left(0.5 + {\left(\frac{0.25}{x}\right)}^{2}\right) - \frac{0.125}{x}}{x}} \]

13. Simplified 98.9%

    \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5} \cdot \frac{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}{x}} \]

14. Taylor expanded in x around 0 7.8%

    \[\leadsto \color{blue}{1} \cdot \frac{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}{x} \]

15. Final simplification 7.8%

    \[\leadsto \frac{0.5 + \frac{\frac{0.0625}{x} + -0.125}{x}}{x} \]
16. Add Preprocessing

Alternative 9: 7.9% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 0.5 x))

C

double code(double x) {
	return 0.5 / x;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 / x;
}

Python

def code(x):
	return 0.5 / x

Julia

function code(x)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / x;
end

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. frac-sub 39.0%

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

   2. div-inv 39.0%

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

   3. *-un-lft-identity 39.0%

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

   4. +-commutative 39.0%

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

   5. *-rgt-identity 39.0%

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

   6. metadata-eval 39.0%

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

   7. frac-times 39.0%

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

   8. associate-*l/ 39.0%

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

   9. *-un-lft-identity 39.0%

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

   10. inv-pow 39.0%

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

   11. sqrt-pow2 39.0%

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

   12. +-commutative 39.0%

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

   13. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. associate-*r/ 39.0%

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

   2. *-rgt-identity 39.0%

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

   3. times-frac 39.0%

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

   4. div-sub 38.9%

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

   5. sub-neg 38.9%

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

   6. *-inverses 38.9%

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

   7. metadata-eval 38.9%

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

   8. /-rgt-identity 38.9%

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

6. Simplified 38.9%

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

7. Taylor expanded in x around inf 97.9%

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

8. Taylor expanded in x around 0 7.8%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
9. Add Preprocessing

Developer Target 1: 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 2024141 
(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))))))

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