2isqrt (example 3.6)

Percentage Accurate: 38.1% → 99.2%

Time: 8.3s

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: 38.1% 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: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.4%

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

   1. associate--l+ 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*r/ 99.4%

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

   5. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around -inf 99.4%

   \[\leadsto \frac{\frac{\color{blue}{0.5 + -1 \cdot \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]
9. Step-by-step derivation

   1. mul-1-neg 99.4%

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

   2. unsub-neg 99.4%

      \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]

   3. mul-1-neg 99.4%

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

   4. unsub-neg 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{\color{blue}{0.125 - \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}}{x}}{x}}{\sqrt{1 + x}} \]

   5. sub-neg 99.4%

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

   6. associate-*r/ 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-neg-frac 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \color{blue}{\frac{-0.0390625}{x}}}{x}}{x}}{x}}{\sqrt{1 + x}} \]

   9. metadata-eval 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{\color{blue}{-0.0390625}}{x}}{x}}{x}}{x}}{\sqrt{1 + x}} \]

10. Simplified 99.4%

    \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]
11. Step-by-step derivation

    1. div-inv 99.4%

       \[\leadsto \color{blue}{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x} \cdot \frac{1}{\sqrt{1 + x}}} \]

    2. pow1/2 99.4%

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

    3. pow-flip 99.4%

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

    4. +-commutative 99.4%

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

    5. metadata-eval 99.4%

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

12. Applied egg-rr 99.4%

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

    1. associate-*l/ 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 2: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 + \frac{\frac{0.0625 + \frac{-0.0390625}{x}}{x} - 0.125}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.4%

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

   1. associate--l+ 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*r/ 99.4%

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

   5. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around -inf 99.4%

   \[\leadsto \frac{\frac{\color{blue}{0.5 + -1 \cdot \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]
9. Step-by-step derivation

   1. mul-1-neg 99.4%

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

   2. unsub-neg 99.4%

      \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]

   3. mul-1-neg 99.4%

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

   4. unsub-neg 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{\color{blue}{0.125 - \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}}{x}}{x}}{\sqrt{1 + x}} \]

   5. sub-neg 99.4%

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

   6. associate-*r/ 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-neg-frac 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \color{blue}{\frac{-0.0390625}{x}}}{x}}{x}}{x}}{\sqrt{1 + x}} \]

   9. metadata-eval 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{\color{blue}{-0.0390625}}{x}}{x}}{x}}{x}}{\sqrt{1 + x}} \]

10. Simplified 99.4%

    \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\frac{0.5 + \frac{\frac{0.0625 + \frac{-0.0390625}{x}}{x} - 0.125}{x}}{x}}{\sqrt{x + 1}} \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125 + \frac{-0.0625}{x}}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.4%

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

   1. associate--l+ 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*r/ 99.4%

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

   5. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around -inf 99.3%

   \[\leadsto \frac{\frac{\color{blue}{0.5 + -1 \cdot \frac{0.125 - 0.0625 \cdot \frac{1}{x}}{x}}}{x}}{\sqrt{1 + x}} \]
9. Step-by-step derivation

   1. mul-1-neg 99.3%

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

   2. unsub-neg 99.3%

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

   3. sub-neg 99.3%

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

   4. associate-*r/ 99.3%

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

   5. metadata-eval 99.3%

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

   6. distribute-neg-frac 99.3%

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

   7. metadata-eval 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \frac{\frac{0.5 - \frac{0.125 + \frac{-0.0625}{x}}{x}}{x}}{\sqrt{x + 1}} \]
12. Add Preprocessing

Alternative 4: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{x}{0.5 + \frac{-0.125}{x}}}}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

double code(double x) {
	return (1.0 / (x / (0.5 + (-0.125 / x)))) / sqrt((x + 1.0));
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / (x / (0.5 + (-0.125 / x)))) / math.sqrt((x + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-*r/ 99.1%

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

   2. metadata-eval 99.1%

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

7. Simplified 99.1%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

9. Applied egg-rr 99.1%

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

    1. unpow-1 99.1%

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

    2. sub-neg 99.1%

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

    3. distribute-neg-frac 99.1%

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

    4. metadata-eval 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

    \[\leadsto \frac{\frac{1}{\frac{x}{0.5 + \frac{-0.125}{x}}}}{\sqrt{x + 1}} \]
13. Add Preprocessing

Alternative 5: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
}

Fortran

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

Java

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

Python

def code(x):
	return ((0.5 - (0.125 / x)) / x) / math.sqrt((x + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-*r/ 99.1%

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

   2. metadata-eval 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{x + 1}} \]
9. Add Preprocessing

Alternative 6: 98.0% 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 40.3%

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

   1. expm1-log1p-u 40.3%

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

   2. expm1-undefine 4.9%

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

   3. inv-pow 4.9%

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

   4. sqrt-pow2 4.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 4.9%

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

4. Applied egg-rr 4.9%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. log1p-undefine 4.9%

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

   2. rem-exp-log 4.9%

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

   3. +-commutative 4.9%

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

   4. associate--l+ 31.9%

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

   5. metadata-eval 31.9%

      \[\leadsto \left({x}^{-0.5} + \color{blue}{0}\right) - \frac{1}{\sqrt{x + 1}} \]

   6. +-rgt-identity 31.9%

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

6. Simplified 31.9%

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

7. Taylor expanded in x around inf 68.7%

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

   1. *-commutative 68.7%

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

9. Simplified 68.7%

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

    1. *-un-lft-identity 68.7%

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

    2. pow-flip 70.3%

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

    3. sqrt-pow1 98.4%

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

    4. metadata-eval 98.4%

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

    5. metadata-eval 98.4%

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

11. Applied egg-rr 98.4%

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

    1. *-lft-identity 98.4%

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

13. Simplified 98.4%

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

14. Final simplification 98.4%

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

Alternative 7: 37.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x} - 0.5}{x}}{-1 - 0.5 \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.4%

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

   1. associate--l+ 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*r/ 99.4%

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

   5. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around -inf 99.4%

   \[\leadsto \frac{\frac{\color{blue}{0.5 + -1 \cdot \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]
9. Step-by-step derivation

   1. mul-1-neg 99.4%

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

   2. unsub-neg 99.4%

      \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]

   3. mul-1-neg 99.4%

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

   4. unsub-neg 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{\color{blue}{0.125 - \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}}{x}}{x}}{\sqrt{1 + x}} \]

   5. sub-neg 99.4%

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

   6. associate-*r/ 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-neg-frac 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \color{blue}{\frac{-0.0390625}{x}}}{x}}{x}}{x}}{\sqrt{1 + x}} \]

   9. metadata-eval 99.4%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{\color{blue}{-0.0390625}}{x}}{x}}{x}}{x}}{\sqrt{1 + x}} \]

10. Simplified 99.4%

    \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}}{x}}{\sqrt{1 + x}} \]

11. Taylor expanded in x around 0 39.8%

    \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}}{\color{blue}{1 + 0.5 \cdot x}} \]
12. Step-by-step derivation

    1. *-commutative 39.8%

       \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}}{1 + \color{blue}{x \cdot 0.5}} \]

13. Simplified 39.8%

    \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}}{\color{blue}{1 + x \cdot 0.5}} \]

14. Final simplification 39.8%

    \[\leadsto \frac{\frac{\frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x} - 0.5}{x}}{-1 - 0.5 \cdot x} \]
15. Add Preprocessing

Alternative 8: 37.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.125}{x} - 0.5}{x}}{-1 - 0.5 \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (((0.125 / x) - 0.5) / x) / (-1.0 - (0.5 * x));
}

Python

def code(x):
	return (((0.125 / x) - 0.5) / x) / (-1.0 - (0.5 * x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. frac-sub 40.4%

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

   2. associate-/r* 40.4%

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

   3. *-un-lft-identity 40.4%

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

   4. *-rgt-identity 40.4%

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

   5. +-commutative 40.4%

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

   6. +-commutative 40.4%

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

4. Applied egg-rr 40.4%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-*r/ 99.1%

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

   2. metadata-eval 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in x around 0 39.8%

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

   1. *-commutative 39.8%

      \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}}{1 + \color{blue}{x \cdot 0.5}} \]

10. Simplified 39.8%

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

11. Final simplification 39.8%

    \[\leadsto \frac{\frac{\frac{0.125}{x} - 0.5}{x}}{-1 - 0.5 \cdot x} \]
12. Add Preprocessing

Alternative 9: 35.3% 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 40.3%

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

   1. add-exp-log 6.7%

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

   2. log-rec 6.7%

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

   3. pow1/2 6.7%

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

   4. log-pow 6.7%

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

   5. +-commutative 6.7%

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

   6. log1p-define 6.7%

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

4. Applied egg-rr 6.7%

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

5. Taylor expanded in x around inf 4.6%

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

   1. distribute-lft-neg-in 4.6%

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

   2. metadata-eval 4.6%

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

   3. *-commutative 4.6%

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

   4. exp-to-pow 37.8%

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

   5. unpow1/2 37.8%

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

   6. +-inverses 37.8%

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

7. Simplified 37.8%

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

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.2% 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 2024182 
(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)))))