2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.2%

Time: 15.8s

Alternatives: 5

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{0.34375 \cdot {x}^{-2} + \frac{-0.75}{x}} \cdot \left(0.5 \cdot {x}^{-1.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (+ (* 0.34375 (pow x -2.0)) (/ -0.75 x))) (* 0.5 (pow x -1.5))))

C

double code(double x) {
	return exp(((0.34375 * pow(x, -2.0)) + (-0.75 / x))) * (0.5 * pow(x, -1.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(((0.34375d0 * (x ** (-2.0d0))) + ((-0.75d0) / x))) * (0.5d0 * (x ** (-1.5d0)))
end function

Java

public static double code(double x) {
	return Math.exp(((0.34375 * Math.pow(x, -2.0)) + (-0.75 / x))) * (0.5 * Math.pow(x, -1.5));
}

Python

def code(x):
	return math.exp(((0.34375 * math.pow(x, -2.0)) + (-0.75 / x))) * (0.5 * math.pow(x, -1.5))

Julia

function code(x)
	return Float64(exp(Float64(Float64(0.34375 * (x ^ -2.0)) + Float64(-0.75 / x))) * Float64(0.5 * (x ^ -1.5)))
end

MATLAB

function tmp = code(x)
	tmp = exp(((0.34375 * (x ^ -2.0)) + (-0.75 / x))) * (0.5 * (x ^ -1.5));
end

Wolfram

code[x_] := N[(N[Exp[N[(N[(0.34375 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.75 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. add-exp-log 38.6%

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

   2. inv-pow 38.6%

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

   3. sqrt-pow2 32.8%

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

   4. metadata-eval 32.8%

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

   5. inv-pow 32.8%

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

   6. sqrt-pow2 38.6%

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

   7. +-commutative 38.6%

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

   8. metadata-eval 38.6%

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

4. Applied egg-rr 38.6%

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

5. Taylor expanded in x around inf 93.1%

   \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\log \left(\frac{1}{x}\right) + 0.34375 \cdot \frac{1}{{x}^{2}}\right)\right) - 0.75 \cdot \frac{1}{x}}} \]
6. Step-by-step derivation

   1. associate--l+ 93.1%

      \[\leadsto e^{\color{blue}{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\log \left(\frac{1}{x}\right) + 0.34375 \cdot \frac{1}{{x}^{2}}\right) - 0.75 \cdot \frac{1}{x}\right)}} \]

   2. distribute-lft-neg-in 93.1%

      \[\leadsto e^{\log \color{blue}{\left(\left(--0.5\right) \cdot \sqrt{\frac{1}{x}}\right)} + \left(\left(\log \left(\frac{1}{x}\right) + 0.34375 \cdot \frac{1}{{x}^{2}}\right) - 0.75 \cdot \frac{1}{x}\right)} \]

   3. metadata-eval 93.1%

      \[\leadsto e^{\log \left(\color{blue}{0.5} \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\log \left(\frac{1}{x}\right) + 0.34375 \cdot \frac{1}{{x}^{2}}\right) - 0.75 \cdot \frac{1}{x}\right)} \]

   4. sub-neg 93.1%

      \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\left(\left(\log \left(\frac{1}{x}\right) + 0.34375 \cdot \frac{1}{{x}^{2}}\right) + \left(-0.75 \cdot \frac{1}{x}\right)\right)}} \]

   5. +-commutative 93.1%

      \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\color{blue}{\left(0.34375 \cdot \frac{1}{{x}^{2}} + \log \left(\frac{1}{x}\right)\right)} + \left(-0.75 \cdot \frac{1}{x}\right)\right)} \]

   6. log-rec 93.1%

      \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(0.34375 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log x\right)}\right) + \left(-0.75 \cdot \frac{1}{x}\right)\right)} \]

   7. unsub-neg 93.1%

      \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\color{blue}{\left(0.34375 \cdot \frac{1}{{x}^{2}} - \log x\right)} + \left(-0.75 \cdot \frac{1}{x}\right)\right)} \]

   8. associate-*r/ 93.1%

      \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\color{blue}{\frac{0.34375 \cdot 1}{{x}^{2}}} - \log x\right) + \left(-0.75 \cdot \frac{1}{x}\right)\right)} \]

   9. metadata-eval 93.1%

      \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\frac{\color{blue}{0.34375}}{{x}^{2}} - \log x\right) + \left(-0.75 \cdot \frac{1}{x}\right)\right)} \]

   10. associate-*r/ 93.1%

       \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\frac{0.34375}{{x}^{2}} - \log x\right) + \left(-\color{blue}{\frac{0.75 \cdot 1}{x}}\right)\right)} \]

   11. metadata-eval 93.1%

       \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\frac{0.34375}{{x}^{2}} - \log x\right) + \left(-\frac{\color{blue}{0.75}}{x}\right)\right)} \]

   12. distribute-neg-frac 93.1%

       \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\frac{0.34375}{{x}^{2}} - \log x\right) + \color{blue}{\frac{-0.75}{x}}\right)} \]

   13. metadata-eval 93.1%

       \[\leadsto e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\frac{0.34375}{{x}^{2}} - \log x\right) + \frac{\color{blue}{-0.75}}{x}\right)} \]

7. Simplified 93.1%

   \[\leadsto e^{\color{blue}{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(\left(\frac{0.34375}{{x}^{2}} - \log x\right) + \frac{-0.75}{x}\right)}} \]

8. Taylor expanded in x around 0 93.1%

   \[\leadsto \color{blue}{e^{\left(\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) + 0.34375 \cdot \frac{1}{{x}^{2}}\right) - \left(\log x + 0.75 \cdot \frac{1}{x}\right)}} \]

10. Simplified 99.1%

    \[\leadsto \color{blue}{e^{\frac{-0.75}{x}} \cdot \left(\left(0.5 \cdot {x}^{-0.5}\right) \cdot \frac{e^{\frac{0.34375}{{x}^{2}}}}{x}\right)} \]
11. Step-by-step derivation

    1. pow1 99.1%

       \[\leadsto \color{blue}{{\left(e^{\frac{-0.75}{x}} \cdot \left(\left(0.5 \cdot {x}^{-0.5}\right) \cdot \frac{e^{\frac{0.34375}{{x}^{2}}}}{x}\right)\right)}^{1}} \]

12. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{{\left(\left({\left(e^{0.34375}\right)}^{\left({x}^{-2}\right)} \cdot \left(0.5 \cdot {x}^{-1.5}\right)\right) \cdot e^{\frac{-0.75}{x}}\right)}^{1}} \]
13. Step-by-step derivation

    1. unpow1 99.5%

       \[\leadsto \color{blue}{\left({\left(e^{0.34375}\right)}^{\left({x}^{-2}\right)} \cdot \left(0.5 \cdot {x}^{-1.5}\right)\right) \cdot e^{\frac{-0.75}{x}}} \]

    2. associate-*l* 99.5%

       \[\leadsto \color{blue}{{\left(e^{0.34375}\right)}^{\left({x}^{-2}\right)} \cdot \left(\left(0.5 \cdot {x}^{-1.5}\right) \cdot e^{\frac{-0.75}{x}}\right)} \]

    3. *-commutative 99.5%

       \[\leadsto {\left(e^{0.34375}\right)}^{\left({x}^{-2}\right)} \cdot \color{blue}{\left(e^{\frac{-0.75}{x}} \cdot \left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

    4. associate-*r* 99.5%

       \[\leadsto \color{blue}{\left({\left(e^{0.34375}\right)}^{\left({x}^{-2}\right)} \cdot e^{\frac{-0.75}{x}}\right) \cdot \left(0.5 \cdot {x}^{-1.5}\right)} \]

    5. exp-prod 99.5%

       \[\leadsto \left(\color{blue}{e^{0.34375 \cdot {x}^{-2}}} \cdot e^{\frac{-0.75}{x}}\right) \cdot \left(0.5 \cdot {x}^{-1.5}\right) \]

    6. prod-exp 99.5%

       \[\leadsto \color{blue}{e^{0.34375 \cdot {x}^{-2} + \frac{-0.75}{x}}} \cdot \left(0.5 \cdot {x}^{-1.5}\right) \]

14. Simplified 99.5%

    \[\leadsto \color{blue}{e^{0.34375 \cdot {x}^{-2} + \frac{-0.75}{x}} \cdot \left(0.5 \cdot {x}^{-1.5}\right)} \]

15. Final simplification 99.5%

    \[\leadsto e^{0.34375 \cdot {x}^{-2} + \frac{-0.75}{x}} \cdot \left(0.5 \cdot {x}^{-1.5}\right) \]
16. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot {x}^{-1.5}\right) \cdot e^{\frac{-0.75}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 0.5 (pow x -1.5)) (exp (/ -0.75 x))))

C

double code(double x) {
	return (0.5 * pow(x, -1.5)) * exp((-0.75 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-1.5d0))) * exp(((-0.75d0) / x))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -1.5)) * Math.exp((-0.75 / x));
}

Python

def code(x):
	return (0.5 * math.pow(x, -1.5)) * math.exp((-0.75 / x))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -1.5)) * exp(Float64(-0.75 / x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -1.5)) * exp((-0.75 / x));
end

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-0.75 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. add-exp-log 38.6%

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

   2. inv-pow 38.6%

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

   3. sqrt-pow2 32.8%

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

   4. metadata-eval 32.8%

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

   5. inv-pow 32.8%

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

   6. sqrt-pow2 38.6%

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

   7. +-commutative 38.6%

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

   8. metadata-eval 38.6%

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

4. Applied egg-rr 38.6%

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

5. Taylor expanded in x around inf 92.8%

   \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \log \left(\frac{1}{x}\right)\right) - 0.75 \cdot \frac{1}{x}}} \]
6. Step-by-step derivation

   1. sub-neg 92.8%

      \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \log \left(\frac{1}{x}\right)\right) + \left(-0.75 \cdot \frac{1}{x}\right)}} \]

   2. log-rec 92.8%

      \[\leadsto e^{\left(\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\left(-\log x\right)}\right) + \left(-0.75 \cdot \frac{1}{x}\right)} \]

   3. unsub-neg 92.8%

      \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x\right)} + \left(-0.75 \cdot \frac{1}{x}\right)} \]

   4. distribute-lft-neg-in 92.8%

      \[\leadsto e^{\left(\log \color{blue}{\left(\left(--0.5\right) \cdot \sqrt{\frac{1}{x}}\right)} - \log x\right) + \left(-0.75 \cdot \frac{1}{x}\right)} \]

   5. metadata-eval 92.8%

      \[\leadsto e^{\left(\log \left(\color{blue}{0.5} \cdot \sqrt{\frac{1}{x}}\right) - \log x\right) + \left(-0.75 \cdot \frac{1}{x}\right)} \]

   6. associate-*r/ 92.8%

      \[\leadsto e^{\left(\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x\right) + \left(-\color{blue}{\frac{0.75 \cdot 1}{x}}\right)} \]

   7. metadata-eval 92.8%

      \[\leadsto e^{\left(\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x\right) + \left(-\frac{\color{blue}{0.75}}{x}\right)} \]

   8. distribute-neg-frac 92.8%

      \[\leadsto e^{\left(\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x\right) + \color{blue}{\frac{-0.75}{x}}} \]

   9. metadata-eval 92.8%

      \[\leadsto e^{\left(\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x\right) + \frac{\color{blue}{-0.75}}{x}} \]

7. Simplified 92.8%

   \[\leadsto e^{\color{blue}{\left(\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x\right) + \frac{-0.75}{x}}} \]
8. Step-by-step derivation

   1. exp-sum 92.7%

      \[\leadsto \color{blue}{e^{\log \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) - \log x} \cdot e^{\frac{-0.75}{x}}} \]

   2. diff-log 93.0%

      \[\leadsto e^{\color{blue}{\log \left(\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right)}} \cdot e^{\frac{-0.75}{x}} \]

   3. add-exp-log 98.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \cdot e^{\frac{-0.75}{x}} \]

   4. inv-pow 98.8%

      \[\leadsto \frac{0.5 \cdot \sqrt{\color{blue}{{x}^{-1}}}}{x} \cdot e^{\frac{-0.75}{x}} \]

   5. sqrt-pow1 98.8%

      \[\leadsto \frac{0.5 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x} \cdot e^{\frac{-0.75}{x}} \]

   6. metadata-eval 98.8%

      \[\leadsto \frac{0.5 \cdot {x}^{\color{blue}{-0.5}}}{x} \cdot e^{\frac{-0.75}{x}} \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x} \cdot e^{\frac{-0.75}{x}}} \]
10. Step-by-step derivation

    1. div-inv 98.7%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot {x}^{-0.5}\right) \cdot \frac{1}{x}\right)} \cdot e^{\frac{-0.75}{x}} \]

    2. associate-*l* 98.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left({x}^{-0.5} \cdot \frac{1}{x}\right)\right)} \cdot e^{\frac{-0.75}{x}} \]

    3. inv-pow 98.7%

       \[\leadsto \left(0.5 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{-1}}\right)\right) \cdot e^{\frac{-0.75}{x}} \]

    4. metadata-eval 98.7%

       \[\leadsto \left(0.5 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right)\right) \cdot e^{\frac{-0.75}{x}} \]

    5. pow-prod-up 98.3%

       \[\leadsto \left(0.5 \cdot \left({x}^{-0.5} \cdot \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)}\right)\right) \cdot e^{\frac{-0.75}{x}} \]

    6. associate-*l* 98.3%

       \[\leadsto \left(0.5 \cdot \color{blue}{\left(\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot {x}^{-0.5}\right)}\right) \cdot e^{\frac{-0.75}{x}} \]

    7. *-commutative 98.3%

       \[\leadsto \color{blue}{\left(\left(\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot {x}^{-0.5}\right) \cdot 0.5\right)} \cdot e^{\frac{-0.75}{x}} \]

    8. associate-*l* 98.3%

       \[\leadsto \left(\color{blue}{\left({x}^{-0.5} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)} \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    9. pow-prod-up 98.7%

       \[\leadsto \left(\left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5 + -0.5\right)}}\right) \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    10. metadata-eval 98.7%

        \[\leadsto \left(\left({x}^{-0.5} \cdot {x}^{\color{blue}{-1}}\right) \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    11. inv-pow 98.7%

        \[\leadsto \left(\left({x}^{-0.5} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    12. div-inv 98.8%

        \[\leadsto \left(\color{blue}{\frac{{x}^{-0.5}}{x}} \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    13. pow1 98.8%

        \[\leadsto \left(\frac{{x}^{-0.5}}{\color{blue}{{x}^{1}}} \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    14. pow-div 99.0%

        \[\leadsto \left(\color{blue}{{x}^{\left(-0.5 - 1\right)}} \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

    15. metadata-eval 99.0%

        \[\leadsto \left({x}^{\color{blue}{-1.5}} \cdot 0.5\right) \cdot e^{\frac{-0.75}{x}} \]

11. Applied egg-rr 99.0%

    \[\leadsto \color{blue}{\left({x}^{-1.5} \cdot 0.5\right)} \cdot e^{\frac{-0.75}{x}} \]

12. Final simplification 99.0%

    \[\leadsto \left(0.5 \cdot {x}^{-1.5}\right) \cdot e^{\frac{-0.75}{x}} \]
13. Add Preprocessing

Alternative 3: 97.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. frac-sub 38.7%

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

   2. div-inv 38.7%

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

   3. *-un-lft-identity 38.7%

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

   4. +-commutative 38.7%

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

   5. *-rgt-identity 38.7%

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

   6. metadata-eval 38.7%

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

   7. frac-times 38.7%

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

   8. associate-*l/ 38.7%

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

   9. *-un-lft-identity 38.7%

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

   10. pow1/2 38.7%

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

   11. pow-flip 38.7%

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

   12. +-commutative 38.7%

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

   13. metadata-eval 38.7%

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

4. Applied egg-rr 38.7%

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

   1. *-commutative 38.7%

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

   2. /-rgt-identity 38.7%

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

   3. times-frac 38.7%

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

   4. *-commutative 38.7%

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

   5. times-frac 38.7%

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

   6. div-sub 38.7%

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

   7. sub-neg 38.7%

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

   8. *-inverses 38.7%

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

   9. metadata-eval 38.7%

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

6. Simplified 38.7%

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

7. Taylor expanded in x around inf 97.8%

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

8. Final simplification 97.8%

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

Alternative 4: 37.6% 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 9.8%

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

      1. frac-2neg 9.8%

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

      2. metadata-eval 9.8%

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

      3. frac-sub 10.0%

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

      4. *-un-lft-identity 10.0%

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

      5. +-commutative 10.0%

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

      6. +-commutative 10.0%

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

   4. Applied egg-rr 10.0%

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

   5. Taylor expanded in x around inf 7.9%

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

      1. mul-1-neg 7.9%

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

   7. Simplified 7.9%

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

   8. Taylor expanded in x around 0 8.1%

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

   if 4.6000000000000003e153 < x

   1. Initial program 66.9%

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

   3. Taylor expanded in x around inf 66.9%

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

4. Final simplification 37.8%

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

Alternative 5: 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 38.6%

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

3. Taylor expanded in x around inf 36.0%

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

4. Final simplification 36.0%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(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)))))