2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.3%

Time: 8.7s

Alternatives: 7

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: 38.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(0.375, {x}^{-2.5}, -0.5 \cdot {x}^{-1.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.3125 (pow x -3.5)) (fma 0.375 (pow x -2.5) (* -0.5 (pow x -1.5)))))

C

double code(double x) {
	return (0.3125 * pow(x, -3.5)) - fma(0.375, pow(x, -2.5), (-0.5 * pow(x, -1.5)));
}

Julia

function code(x)
	return Float64(Float64(0.3125 * (x ^ -3.5)) - fma(0.375, (x ^ -2.5), Float64(-0.5 * (x ^ -1.5))))
end

Wolfram

code[x_] := N[(N[(0.3125 * N[Power[x, -3.5], $MachinePrecision]), $MachinePrecision] - N[(0.375 * N[Power[x, -2.5], $MachinePrecision] + N[(-0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. add-exp-log 6.8%

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

   2. log-rec 6.8%

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

   3. pow1/2 6.8%

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

   4. log-pow 6.8%

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

   5. +-commutative 6.8%

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

   6. log1p-define 6.8%

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

4. Applied egg-rr 6.8%

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

5. Taylor expanded in x around inf 6.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} + 0.3125 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{3}}\right) - \left(e^{--0.5 \cdot \log \left(\frac{1}{x}\right)} + \left(-0.5 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{x} + 0.375 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 6.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\frac{1}{x}} + 0.3125 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{3}}\right) - e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}\right) - \left(-0.5 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{x} + 0.375 \cdot \frac{e^{--0.5 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{2}}\right)} \]

7. Simplified 98.4%

   \[\leadsto \color{blue}{\left(\frac{0.3125 \cdot {x}^{-0.5}}{{x}^{3}} + 0\right) - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)} \]
8. Step-by-step derivation

   1. +-lft-identity 98.4%

      \[\leadsto \color{blue}{\left(0 + \left(\frac{0.3125 \cdot {x}^{-0.5}}{{x}^{3}} + 0\right)\right)} - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right) \]

   2. associate--l+ 98.4%

      \[\leadsto \color{blue}{0 + \left(\left(\frac{0.3125 \cdot {x}^{-0.5}}{{x}^{3}} + 0\right) - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right)} \]

   3. +-rgt-identity 98.4%

      \[\leadsto 0 + \left(\color{blue}{\frac{0.3125 \cdot {x}^{-0.5}}{{x}^{3}}} - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right) \]

   4. associate-/l* 98.4%

      \[\leadsto 0 + \left(\color{blue}{0.3125 \cdot \frac{{x}^{-0.5}}{{x}^{3}}} - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right) \]

   5. pow-div 98.4%

      \[\leadsto 0 + \left(0.3125 \cdot \color{blue}{{x}^{\left(-0.5 - 3\right)}} - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right) \]

   6. metadata-eval 98.4%

      \[\leadsto 0 + \left(0.3125 \cdot {x}^{\color{blue}{-3.5}} - \mathsf{fma}\left(-0.5, {\left({x}^{-0.5}\right)}^{3}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right) \]

   7. pow-pow 99.1%

      \[\leadsto 0 + \left(0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right) \]

   8. metadata-eval 99.1%

      \[\leadsto 0 + \left(0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, {x}^{\color{blue}{-1.5}}, \frac{{x}^{-0.5}}{{x}^{2}} \cdot 0.375\right)\right) \]

   9. pow-div 99.1%

      \[\leadsto 0 + \left(0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, {x}^{-1.5}, \color{blue}{{x}^{\left(-0.5 - 2\right)}} \cdot 0.375\right)\right) \]

   10. metadata-eval 99.1%

       \[\leadsto 0 + \left(0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, {x}^{-1.5}, {x}^{\color{blue}{-2.5}} \cdot 0.375\right)\right) \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{0 + \left(0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, {x}^{-1.5}, {x}^{-2.5} \cdot 0.375\right)\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 99.1%

       \[\leadsto \color{blue}{0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(-0.5, {x}^{-1.5}, {x}^{-2.5} \cdot 0.375\right)} \]

    2. fma-define 99.1%

       \[\leadsto 0.3125 \cdot {x}^{-3.5} - \color{blue}{\left(-0.5 \cdot {x}^{-1.5} + {x}^{-2.5} \cdot 0.375\right)} \]

    3. +-commutative 99.1%

       \[\leadsto 0.3125 \cdot {x}^{-3.5} - \color{blue}{\left({x}^{-2.5} \cdot 0.375 + -0.5 \cdot {x}^{-1.5}\right)} \]

    4. *-commutative 99.1%

       \[\leadsto 0.3125 \cdot {x}^{-3.5} - \left(\color{blue}{0.375 \cdot {x}^{-2.5}} + -0.5 \cdot {x}^{-1.5}\right) \]

    5. fma-define 99.1%

       \[\leadsto 0.3125 \cdot {x}^{-3.5} - \color{blue}{\mathsf{fma}\left(0.375, {x}^{-2.5}, -0.5 \cdot {x}^{-1.5}\right)} \]

11. Simplified 99.1%

    \[\leadsto \color{blue}{0.3125 \cdot {x}^{-3.5} - \mathsf{fma}\left(0.375, {x}^{-2.5}, -0.5 \cdot {x}^{-1.5}\right)} \]
12. Add Preprocessing

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) + \left(0.125 \cdot \frac{-1}{x} + 0.0390625 \cdot \frac{-1}{{x}^{3}}\right)}{x} \cdot {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    (+ 0.5 (/ 0.0625 (pow x 2.0)))
    (+ (* 0.125 (/ -1.0 x)) (* 0.0390625 (/ -1.0 (pow x 3.0)))))
   x)
  (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return (((0.5 + (0.0625 / pow(x, 2.0))) + ((0.125 * (-1.0 / x)) + (0.0390625 * (-1.0 / pow(x, 3.0))))) / x) * pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.5d0 + (0.0625d0 / (x ** 2.0d0))) + ((0.125d0 * ((-1.0d0) / x)) + (0.0390625d0 * ((-1.0d0) / (x ** 3.0d0))))) / x) * ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return (((0.5 + (0.0625 / Math.pow(x, 2.0))) + ((0.125 * (-1.0 / x)) + (0.0390625 * (-1.0 / Math.pow(x, 3.0))))) / x) * Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return (((0.5 + (0.0625 / math.pow(x, 2.0))) + ((0.125 * (-1.0 / x)) + (0.0390625 * (-1.0 / math.pow(x, 3.0))))) / x) * math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.0625 / (x ^ 2.0))) + Float64(Float64(0.125 * Float64(-1.0 / x)) + Float64(0.0390625 * Float64(-1.0 / (x ^ 3.0))))) / x) * (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 + (0.0625 / (x ^ 2.0))) + ((0.125 * (-1.0 / x)) + (0.0390625 * (-1.0 / (x ^ 3.0))))) / x) * ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 + N[(0.0625 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.125 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(0.0390625 * N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. frac-sub 39.4%

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

   2. div-inv 39.4%

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

   3. *-un-lft-identity 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. +-commutative 39.4%

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

   6. metadata-eval 39.4%

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

   7. frac-times 39.4%

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

   8. associate-*l/ 39.4%

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

   9. *-un-lft-identity 39.4%

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

   10. inv-pow 39.4%

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

   11. sqrt-pow2 39.4%

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

   12. +-commutative 39.4%

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

   13. metadata-eval 39.4%

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

4. Applied egg-rr 39.4%

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

   1. associate-*r/ 39.4%

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

   2. *-rgt-identity 39.4%

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

   3. times-frac 39.4%

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

   4. div-sub 39.4%

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

   5. sub-neg 39.4%

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

   6. *-inverses 39.4%

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

   7. metadata-eval 39.4%

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

   8. /-rgt-identity 39.4%

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

6. Simplified 39.4%

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

7. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \left(0.125 \cdot \frac{1}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]

8. Final simplification 99.0%

   \[\leadsto \frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) + \left(0.125 \cdot \frac{-1}{x} + 0.0390625 \cdot \frac{-1}{{x}^{3}}\right)}{x} \cdot {\left(x + 1\right)}^{-0.5} \]
9. Add Preprocessing

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{-0.5} \cdot \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)\right) - \frac{0.0390625}{{x}^{3}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (+ x 1.0) -0.5)
  (/
   (-
    (+ (/ 0.0625 (pow x 2.0)) (+ 0.5 (/ -0.125 x)))
    (/ 0.0390625 (pow x 3.0)))
   x)))

C

double code(double x) {
	return pow((x + 1.0), -0.5) * ((((0.0625 / pow(x, 2.0)) + (0.5 + (-0.125 / x))) - (0.0390625 / pow(x, 3.0))) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) * ((((0.0625d0 / (x ** 2.0d0)) + (0.5d0 + ((-0.125d0) / x))) - (0.0390625d0 / (x ** 3.0d0))) / x)
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) * ((((0.0625 / Math.pow(x, 2.0)) + (0.5 + (-0.125 / x))) - (0.0390625 / Math.pow(x, 3.0))) / x);
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) * ((((0.0625 / math.pow(x, 2.0)) + (0.5 + (-0.125 / x))) - (0.0390625 / math.pow(x, 3.0))) / x)

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) * Float64(Float64(Float64(Float64(0.0625 / (x ^ 2.0)) + Float64(0.5 + Float64(-0.125 / x))) - Float64(0.0390625 / (x ^ 3.0))) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) * ((((0.0625 / (x ^ 2.0)) + (0.5 + (-0.125 / x))) - (0.0390625 / (x ^ 3.0))) / x);
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(N[(N[(0.0625 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0390625 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. frac-sub 39.4%

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

   2. div-inv 39.4%

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

   3. *-un-lft-identity 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. +-commutative 39.4%

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

   6. metadata-eval 39.4%

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

   7. frac-times 39.4%

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

   8. associate-*l/ 39.4%

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

   9. *-un-lft-identity 39.4%

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

   10. inv-pow 39.4%

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

   11. sqrt-pow2 39.4%

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

   12. +-commutative 39.4%

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

   13. metadata-eval 39.4%

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

4. Applied egg-rr 39.4%

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

   1. associate-*r/ 39.4%

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

   2. *-rgt-identity 39.4%

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

   3. times-frac 39.4%

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

   4. div-sub 39.4%

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

   5. sub-neg 39.4%

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

   6. *-inverses 39.4%

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

   7. metadata-eval 39.4%

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

   8. /-rgt-identity 39.4%

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

6. Simplified 39.4%

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

7. Taylor expanded in x around inf 99.0%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \left(0.125 \cdot \frac{1}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate--r+ 99.0%

      \[\leadsto \frac{\color{blue}{\left(\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. +-commutative 99.0%

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{0.0625}{{x}^{2}} + 0.5\right)} - 0.125 \cdot \frac{1}{x}\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   3. associate--l+ 99.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 - 0.125 \cdot \frac{1}{x}\right)\right)} - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   4. sub-neg 99.0%

      \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \color{blue}{\left(0.5 + \left(-0.125 \cdot \frac{1}{x}\right)\right)}\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   5. associate-*r/ 99.0%

      \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \left(-\color{blue}{\frac{0.125 \cdot 1}{x}}\right)\right)\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   6. metadata-eval 99.0%

      \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \left(-\frac{\color{blue}{0.125}}{x}\right)\right)\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   7. distribute-neg-frac 99.0%

      \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \color{blue}{\frac{-0.125}{x}}\right)\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   8. metadata-eval 99.0%

      \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{\color{blue}{-0.125}}{x}\right)\right) - 0.0390625 \cdot \frac{1}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   9. associate-*r/ 99.0%

      \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)\right) - \color{blue}{\frac{0.0390625 \cdot 1}{{x}^{3}}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   10. metadata-eval 99.0%

       \[\leadsto \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)\right) - \frac{\color{blue}{0.0390625}}{{x}^{3}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)\right) - \frac{0.0390625}{{x}^{3}}}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Final simplification 99.0%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \frac{\left(\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)\right) - \frac{0.0390625}{{x}^{3}}}{x} \]
11. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{-0.5} \cdot \frac{\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (+ x 1.0) -0.5)
  (/ (+ (/ 0.0625 (pow x 2.0)) (+ 0.5 (/ -0.125 x))) x)))

C

double code(double x) {
	return pow((x + 1.0), -0.5) * (((0.0625 / pow(x, 2.0)) + (0.5 + (-0.125 / x))) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) * (((0.0625d0 / (x ** 2.0d0)) + (0.5d0 + ((-0.125d0) / x))) / x)
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) * (((0.0625 / Math.pow(x, 2.0)) + (0.5 + (-0.125 / x))) / x);
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) * (((0.0625 / math.pow(x, 2.0)) + (0.5 + (-0.125 / x))) / x)

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) * Float64(Float64(Float64(0.0625 / (x ^ 2.0)) + Float64(0.5 + Float64(-0.125 / x))) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) * (((0.0625 / (x ^ 2.0)) + (0.5 + (-0.125 / x))) / x);
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(N[(0.0625 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. frac-sub 39.4%

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

   2. div-inv 39.4%

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

   3. *-un-lft-identity 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. +-commutative 39.4%

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

   6. metadata-eval 39.4%

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

   7. frac-times 39.4%

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

   8. associate-*l/ 39.4%

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

   9. *-un-lft-identity 39.4%

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

   10. inv-pow 39.4%

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

   11. sqrt-pow2 39.4%

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

   12. +-commutative 39.4%

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

   13. metadata-eval 39.4%

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

4. Applied egg-rr 39.4%

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

   1. associate-*r/ 39.4%

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

   2. *-rgt-identity 39.4%

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

   3. times-frac 39.4%

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

   4. div-sub 39.4%

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

   5. sub-neg 39.4%

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

   6. *-inverses 39.4%

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

   7. metadata-eval 39.4%

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

   8. /-rgt-identity 39.4%

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

6. Simplified 39.4%

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

7. Taylor expanded in x around inf 98.8%

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

   1. +-commutative 98.8%

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

   2. associate--l+ 98.9%

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

   3. sub-neg 98.9%

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

   4. associate-*r/ 98.9%

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

   5. metadata-eval 98.9%

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

   6. distribute-neg-frac 98.9%

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

   7. metadata-eval 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \frac{\frac{0.0625}{{x}^{2}} + \left(0.5 + \frac{-0.125}{x}\right)}{x} \]
11. Add Preprocessing

Alternative 5: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.4%

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

   1. frac-sub 39.4%

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

   2. div-inv 39.4%

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

   3. *-un-lft-identity 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. +-commutative 39.4%

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

   6. metadata-eval 39.4%

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

   7. frac-times 39.4%

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

   8. associate-*l/ 39.4%

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

   9. *-un-lft-identity 39.4%

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

   10. inv-pow 39.4%

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

   11. sqrt-pow2 39.4%

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

   12. +-commutative 39.4%

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

   13. metadata-eval 39.4%

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

4. Applied egg-rr 39.4%

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

   1. associate-*r/ 39.4%

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

   2. *-rgt-identity 39.4%

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

   3. times-frac 39.4%

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

   4. div-sub 39.4%

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

   5. sub-neg 39.4%

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

   6. *-inverses 39.4%

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

   7. metadata-eval 39.4%

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

   8. /-rgt-identity 39.4%

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

6. Simplified 39.4%

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

7. Taylor expanded in x around inf 98.6%

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

   1. associate-*r/ 98.6%

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

   2. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 6: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.4%

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

   1. add-exp-log 6.8%

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

   2. log-rec 6.8%

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

   3. pow1/2 6.8%

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

   4. log-pow 6.8%

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

   5. +-commutative 6.8%

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

   6. log1p-define 6.8%

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

4. Applied egg-rr 6.8%

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

5. Taylor expanded in x around inf 5.7%

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

   1. +-commutative 5.7%

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

   2. distribute-lft-neg-in 5.7%

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

   3. metadata-eval 5.7%

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

   4. *-commutative 5.7%

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

   5. exp-to-pow 38.3%

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

   6. unpow1/2 38.3%

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

   7. associate--l+ 93.6%

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

7. Simplified 97.4%

   \[\leadsto \color{blue}{0.5 \cdot {\left({x}^{-0.5}\right)}^{3} + 0} \]
8. Step-by-step derivation

   1. +-rgt-identity 97.4%

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

   2. *-commutative 97.4%

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

   3. pow-pow 98.1%

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

   4. metadata-eval 98.1%

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

9. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
10. Add Preprocessing

Alternative 7: 35.4% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 39.4%

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

   1. add-exp-log 6.8%

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

   2. log-rec 6.8%

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

   3. pow1/2 6.8%

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

   4. log-pow 6.8%

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

   5. +-commutative 6.8%

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

   6. log1p-define 6.8%

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

4. Applied egg-rr 6.8%

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

5. Taylor expanded in x around inf 4.5%

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

   1. distribute-lft-neg-in 4.5%

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

   2. metadata-eval 4.5%

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

   3. *-commutative 4.5%

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

   4. exp-to-pow 36.9%

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

   5. unpow1/2 36.9%

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

   6. +-inverses 36.9%

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

7. Simplified 36.9%

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

Developer Target 2: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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