2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.1%

Time: 23.6s

Alternatives: 6

Speedup: 1.9×

• could not determine a ground truth

  1. x = 2.5385065102351384e+88

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: 38.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. frac-sub 40.0%

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

   2. div-inv 40.0%

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

   3. *-un-lft-identity 40.0%

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

   4. +-commutative 40.0%

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

   5. *-rgt-identity 40.0%

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

   6. metadata-eval 40.0%

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

   7. frac-times 40.0%

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

   8. associate-*l/ 40.0%

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

   9. *-un-lft-identity 40.0%

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

   10. pow1/2 40.0%

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

   11. pow-flip 40.0%

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

   12. +-commutative 40.0%

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

   13. metadata-eval 40.0%

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

4. Applied egg-rr 40.0%

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

   1. *-commutative 40.0%

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

   2. /-rgt-identity 40.0%

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

   3. times-frac 40.0%

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

   4. *-commutative 40.0%

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

   5. times-frac 40.0%

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

   6. div-sub 40.0%

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

   7. sub-neg 40.0%

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

   8. *-inverses 40.0%

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

   9. metadata-eval 40.0%

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

6. Simplified 40.0%

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

   1. /-rgt-identity 40.0%

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

   2. *-commutative 40.0%

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

   3. add-exp-log 40.0%

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

   4. log-prod 40.0%

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

   5. log-pow 40.0%

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

   6. log1p-define 40.0%

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

   7. sqrt-undiv 40.0%

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

8. Applied egg-rr 40.0%

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

   1. prod-exp 40.0%

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

   2. *-commutative 40.0%

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

   3. log1p-undefine 40.0%

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

   4. exp-to-pow 40.0%

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

   5. rem-exp-log 40.0%

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

   6. +-commutative 40.0%

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

10. Simplified 40.0%

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

11. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 99.0%

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

    2. metadata-eval 99.0%

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

    3. associate-*r/ 99.0%

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

    4. metadata-eval 99.0%

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

    5. associate-*r/ 99.0%

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

    6. metadata-eval 99.0%

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

13. Simplified 99.0%

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

14. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. frac-sub 40.0%

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

   2. div-inv 40.0%

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

   3. *-un-lft-identity 40.0%

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

   4. +-commutative 40.0%

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

   5. *-rgt-identity 40.0%

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

   6. metadata-eval 40.0%

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

   7. frac-times 40.0%

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

   8. associate-*l/ 40.0%

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

   9. *-un-lft-identity 40.0%

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

   10. pow1/2 40.0%

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

   11. pow-flip 40.0%

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

   12. +-commutative 40.0%

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

   13. metadata-eval 40.0%

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

4. Applied egg-rr 40.0%

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

   1. *-commutative 40.0%

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

   2. /-rgt-identity 40.0%

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

   3. times-frac 40.0%

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

   4. *-commutative 40.0%

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

   5. times-frac 40.0%

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

   6. div-sub 40.0%

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

   7. sub-neg 40.0%

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

   8. *-inverses 40.0%

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

   9. metadata-eval 40.0%

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

6. Simplified 40.0%

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

   1. /-rgt-identity 40.0%

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

   2. *-commutative 40.0%

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

   3. add-exp-log 40.0%

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

   4. log-prod 40.0%

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

   5. log-pow 40.0%

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

   6. log1p-define 40.0%

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

   7. sqrt-undiv 40.0%

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

8. Applied egg-rr 40.0%

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

   1. prod-exp 40.0%

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

   2. *-commutative 40.0%

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

   3. log1p-undefine 40.0%

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

   4. exp-to-pow 40.0%

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

   5. rem-exp-log 40.0%

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

   6. +-commutative 40.0%

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

10. Simplified 40.0%

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

11. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 99.0%

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

    2. metadata-eval 99.0%

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

    3. associate-*r/ 99.0%

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

    4. metadata-eval 99.0%

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

    5. associate-*r/ 99.0%

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

    6. metadata-eval 99.0%

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

13. Simplified 99.0%

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

14. Taylor expanded in x around inf 98.7%

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

    1. sub-neg 98.7%

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

    2. associate-*r/ 98.7%

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

    3. metadata-eval 98.7%

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

    4. associate-*r/ 98.7%

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

    5. metadata-eval 98.7%

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

    6. distribute-neg-frac 98.7%

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

    7. metadata-eval 98.7%

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

16. Simplified 98.7%

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

17. Final simplification 98.7%

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

Alternative 3: 37.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+153) (/ 1.0 (+ x (sqrt x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 9.9%

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

      1. flip-- 10.1%

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

      2. div-inv 10.1%

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

      3. frac-times 10.2%

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

      4. metadata-eval 10.2%

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

      5. add-sqr-sqrt 9.9%

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

      6. frac-times 9.8%

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

      7. metadata-eval 9.8%

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

      8. add-sqr-sqrt 10.1%

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

      9. +-commutative 10.1%

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

      10. inv-pow 10.1%

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

      11. sqrt-pow2 10.1%

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

      12. metadata-eval 10.1%

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

      13. pow1/2 10.1%

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

      14. pow-flip 10.1%

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

      15. +-commutative 10.1%

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

      16. metadata-eval 10.1%

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

   4. Applied egg-rr 10.1%

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

      1. frac-2neg 10.1%

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

      2. metadata-eval 10.1%

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

      3. frac-sub 13.0%

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

      4. *-un-lft-identity 13.0%

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

      5. distribute-neg-in 13.0%

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

      6. metadata-eval 13.0%

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

      7. distribute-neg-in 13.0%

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

      8. metadata-eval 13.0%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in x around 0 8.2%

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

      1. distribute-rgt-in 8.2%

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

      2. *-lft-identity 8.2%

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

      3. pow-plus 8.2%

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

      4. metadata-eval 8.2%

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

      5. unpow1/2 8.2%

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

   9. Simplified 8.2%

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

   if 4.6000000000000003e153 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in x around inf 67.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. frac-sub 40.0%

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

   2. div-inv 40.0%

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

   3. *-un-lft-identity 40.0%

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

   4. +-commutative 40.0%

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

   5. *-rgt-identity 40.0%

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

   6. metadata-eval 40.0%

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

   7. frac-times 40.0%

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

   8. associate-*l/ 40.0%

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

   9. *-un-lft-identity 40.0%

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

   10. pow1/2 40.0%

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

   11. pow-flip 40.0%

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

   12. +-commutative 40.0%

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

   13. metadata-eval 40.0%

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

4. Applied egg-rr 40.0%

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

   1. *-commutative 40.0%

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

   2. /-rgt-identity 40.0%

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

   3. times-frac 40.0%

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

   4. *-commutative 40.0%

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

   5. times-frac 40.0%

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

   6. div-sub 40.0%

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

   7. sub-neg 40.0%

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

   8. *-inverses 40.0%

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

   9. metadata-eval 40.0%

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

6. Simplified 40.0%

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

   1. /-rgt-identity 40.0%

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

   2. *-commutative 40.0%

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

   3. add-exp-log 40.0%

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

   4. log-prod 40.0%

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

   5. log-pow 40.0%

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

   6. log1p-define 40.0%

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

   7. sqrt-undiv 40.0%

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

8. Applied egg-rr 40.0%

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

   1. prod-exp 40.0%

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

   2. *-commutative 40.0%

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

   3. log1p-undefine 40.0%

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

   4. exp-to-pow 40.0%

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

   5. rem-exp-log 40.0%

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

   6. +-commutative 40.0%

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

10. Simplified 40.0%

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

11. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 99.0%

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

    2. metadata-eval 99.0%

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

    3. associate-*r/ 99.0%

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

    4. metadata-eval 99.0%

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

    5. associate-*r/ 99.0%

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

    6. metadata-eval 99.0%

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

13. Simplified 99.0%

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

14. Taylor expanded in x around inf 97.8%

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

15. Final simplification 97.8%

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

Alternative 5: 37.5% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 4.6e+153) (/ 1.0 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[(1.0 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 9.9%

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

      1. flip-- 10.1%

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

      2. div-inv 10.1%

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

      3. frac-times 10.2%

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

      4. metadata-eval 10.2%

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

      5. add-sqr-sqrt 9.9%

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

      6. frac-times 9.8%

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

      7. metadata-eval 9.8%

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

      8. add-sqr-sqrt 10.1%

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

      9. +-commutative 10.1%

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

      10. inv-pow 10.1%

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

      11. sqrt-pow2 10.1%

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

      12. metadata-eval 10.1%

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

      13. pow1/2 10.1%

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

      14. pow-flip 10.1%

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

      15. +-commutative 10.1%

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

      16. metadata-eval 10.1%

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

   4. Applied egg-rr 10.1%

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

   5. Taylor expanded in x around 0 8.2%

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

      1. distribute-rgt-in 8.2%

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

      2. *-lft-identity 8.2%

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

      3. pow-plus 8.2%

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

      4. metadata-eval 8.2%

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

   7. Simplified 8.2%

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

   8. Taylor expanded in x around inf 8.2%

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

   if 4.6000000000000003e153 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in x around inf 67.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 37.4%

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

4. Final simplification 37.4%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 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]

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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